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{{Short description|Rational number sequence}}
In ], the '''Bernoulli numbers''' are a ] of ]s with deep connections to ]. They are closely related to the values of the ] at negative integers.
{{Use American English|date = March 2019}}
{{Use shortened footnotes|date=April 2021}}
{| class=wikitable style="text-align: right; float:right; clear:right; margin-left:1em;"
|+ Bernoulli numbers {{math|''B''{{su|p=±|b=''n''}}}}
|-
! {{mvar|n}} !! fraction !! decimal
|- decimal
| 0 || 1 || +1.000000000
|-
| 1 || ±{{sfrac|1|2}} || ±0.500000000
|-
| 2 || {{sfrac|1|6}} || +0.166666666
|- style="background:#ABE"
| 3 || 0 || +0.000000000
|-
| 4 || −{{sfrac|1|30}} || −0.033333333
|- style="background:#ABE"
| 5 || 0 || +0.000000000
|-
| 6 || {{sfrac|1|42}} || +0.023809523
|- style="background:#ABE"
| 7 || 0 || +0.000000000
|-
| 8 || −{{sfrac|1|30}} || −0.033333333
|- style="background:#ABE"
| 9 || 0 || +0.000000000
|-
| 10 || {{sfrac|5|66}} || +0.075757575
|- style="background:#ABE"
| 11 || 0 || +0.000000000
|-
| 12 || −{{sfrac|691|2730}} || −0.253113553
|- style="background:#ABE"
| 13 || 0 || +0.000000000
|-
| 14 || {{sfrac|7|6}} || +1.166666666
|- style="background:#ABE"
| 15 || 0 || +0.000000000
|-
| 16 || −{{sfrac|3617|510}} || −7.092156862
|- style="background:#ABE"
| 17 || 0 || +0.000000000
|-
| 18 || {{sfrac|43867|798}} || +54.97117794
|- style="background:#ABE"
| 19 || 0 || +0.000000000
|-
| 20 || −{{sfrac|174611|330}} || −529.1242424
|}


In Europe, they were first studied by ], after whom they were named by ]. In Japan, perhaps earlier, they were independently discovered by ]. They appear in the ] expansions of the ] and ] functions, in the ], and in expressions for certain values of the Riemann zeta function. In ], the '''Bernoulli numbers''' {{math|''B''<sub>''n''</sub>}} are a ] of ]s which occur frequently in ]. The Bernoulli numbers appear in (and can be defined by) the ] expansions of the ] and ] functions, in ] for the sum of ''m''-th powers of the first ''n'' positive integers, in the ], and in expressions for certain values of the ].


The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by <math>B^{-{}}_n</math> and <math>B^{+{}}_n</math>; they differ only for {{math|''n'' {{=}} 1}}, where <math>B^{-{}}_1=-1/2</math> and <math>B^{+{}}_1=+1/2</math>. For every odd {{math|''n'' > 1}}, {{math|''B''<sub>''n''</sub> {{=}} 0}}. For every even {{math|''n'' > 0}}, {{math|''B''<sub>''n''</sub>}} is negative if {{math|''n''}} is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the ] <math>B_n(x)</math>, with <math>B^{-{}}_n=B_n(0)</math> and <math>B^+_n=B_n(1)</math>.{{r|Weisstein2016}}
In ] of ] notes on the ] from 1842, Lovelace describes an ] for generating Bernoulli numbers with ] machine <ref group="~">''Note G'' in the Menabrea reference</ref>. As a result, the Bernoulli numbers have the distinction of being the subject of the first ].


The Bernoulli numbers were discovered around the same time by the Swiss mathematician ], after whom they are named, and independently by Japanese mathematician ]. Seki's discovery was posthumously published in 1712{{r|Selin1997_891|SmithMikami1914_108|Kitagawa}} in his work ''Katsuyō Sanpō''; Bernoulli's, also posthumously, in his '']'' of 1713. ]'s ] on the ] from 1842 describes an ] for generating Bernoulli numbers with ]'s machine;{{r|Menabrea1842_noteG}} it is disputed ]. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex ].
==Introduction==
The Bernoulli numbers ''B''<sub>''n''</sub>, sometimes denoted ''b''<sub>''n''</sub>. were first discovered in connection with the ] of the sums


==Notation==
:<math>\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n </math>
The superscript {{math|±}} used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the {{math|''n'' {{=}} 1}} term is affected:
* {{math|''B''{{su|p=−|b=''n''}} }} with {{math|''B''{{su|p=−|b=1}} {{=}} −{{sfrac|1|2}} }} ({{OEIS2C|id=A027641}} / {{OEIS2C|id=A027642}}) is the sign convention prescribed by ] and most modern textbooks.{{sfnp|Arfken|1970|p=278}}
* {{math|''B''{{su|p=+|b=''n''}}}} with {{math|''B''{{su|p=+|b=1}} {{=}} +{{sfrac|1|2}} }} ({{OEIS2C|id=A164555}} / {{OEIS2C|id=A027642}}) was used in the older literature,{{r|Weisstein2016}} and (since 2022) by ]<ref>] (2022), . {{blockquote|But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half. He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used. It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.
}}</ref> following Peter Luschny's "Bernoulli Manifesto".<ref>Peter Luschny (2013), </ref>


In the formulas below, one can switch from one sign convention to the other with the relation <math>B_n^{+}=(-1)^n B_n^{-}</math>, or for integer {{mvar|n}} = 2 or greater, simply ignore it.
for various fixed values of ''n''. The closed forms are always ]s in ''m'' of degree ''n''&nbsp;+&nbsp;1. The ]s of these polynomials are closely related to the Bernoulli numbers, in connection with ]:


Since {{math|''B''{{sub|''n''}} {{=}} 0}} for all odd {{math|''n'' > 1}}, and many formulas only involve even-index Bernoulli numbers, a few authors write "{{math|''B''{{sub|''n''}}}}" instead of {{math|''B''{{sub|2''n''}}&nbsp;}}. This article does not follow that notation.
:<math>\sum_{k=0}^{m-1} k^n = {1\over{n+1}}\sum_{k=0}^n{n+1\choose{k}} B_k m^{n+1-k}.</math>


==History==
For example, taking ''n'' to be 1,
=== Early history ===
The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.


]
:<math> 0 + 1 + 2 + \cdots + (m-1) = \frac{1}{2}\left(B_0 m^2+2B_1 m^1\right) = \frac{1}{2}\left(m^2-m\right).</math>


Methods to calculate the sum of the first {{mvar|n}} positive integers, the sum of the squares and of the cubes of the first {{mvar|n}} positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were ] (c. 572–497&nbsp;BCE, Greece), ] (287–212&nbsp;BCE, Italy), ] (b. 476, India), ] (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn ] (965–1039, Iraq).
===Generating functions===
The Bernoulli numbers may also be defined using ]s. Their ] is ''x''/(''e<sup>x</sup>''&nbsp;−&nbsp;1), so that:


During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West ] (1560–1621) of England, ] (1580–1635) of Germany, ] (1601–1665) and fellow French mathematician ] (1623–1662) all played important roles.
:<math>
\frac{x}{e^x-1} = \sum_{n=0}^{\infin} B_n \frac{x^n}{n!}
</math>


Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 ''Academia Algebrae'', far higher than anyone before him, but he did not give a general formula.
for all values of ''x'' of ] less than 2π (the ] of this ]).


Blaise Pascal in 1654 proved ] relating {{math|(''n''+1)<sup>''k''+1</sup>}} to the sums of the {{math|''p''}}th powers of the first {{math|''n''}} positive integers for {{math|''p'' {{=}} 0, 1, 2, ..., ''k''}}.
These definitions can be shown to be equivalent using ]. The initial condition <math>B_0 = 1</math> is immediate from ]. To obtain the recurrence, multiply both sides of the equation by <math>e^x-1</math>. Then, using the ] for the ],


The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants {{math|''B''<sub>0</sub>, ''B''<sub>1</sub>, ''B''<sub>2</sub>,...}} which provide a uniform formula for all sums of powers.{{sfnp|Knuth|1993}}
: <math>x = \left( \sum_{j=1}^{\infty} \frac{x^j}{j!} \right) \left( \sum_{k=0}^{\infty} \frac{B_k x^k}{k!} \right). </math>


The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the {{mvar|c}}th powers for any positive integer {{math|''c''}} can be seen from his comment. He wrote:
By expanding this as a ] and rearranging slightly, one obtains


:"With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500."
: <math> x = \sum_{m=0}^{\infty} \left( \sum_{j=0}^{m} {m+1 \choose j} B_j \right) \frac{x^{m+1}}{(m+1)!}. </math>


Bernoulli's result was published posthumously in '']'' in 1713. ] independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.{{r|Selin1997_891}} However, Seki did not present his method as a formula based on a sequence of constants.
It is clear from this last equality that the coefficients in this power series satisfy the same recurrence as the Bernoulli numbers.


Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of ].
=== Other forms and conventions ===


Bernoulli's formula is sometimes called ] after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth{{sfnp|Knuth|1993}} a rigorous proof of Faulhaber's formula was first published by ] in 1834.{{r|Jacobi1834}} Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on):
One may also write


:''"Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants'' {{math|''B''<sub>0</sub>, ''B''<sub>1</sub>, ''B''<sub>2</sub>,}} ''... would provide a uniform''
:<math>\sum_{k=0}^{m-1} k^n = \frac{B_{n+1}(m)-B_{n+1}(0)}{n+1},</math>
::<math display=inline>\sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1} 1 B_1 n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right) </math>
:''for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for'' {{math|Σ ''n<sup>m</sup>''}} ''from polynomials in {{mvar|N}} to polynomials in {{mvar|n}}."{{sfnp|Knuth|1993|p=14}}''


In the above Knuth meant <math>B_1^-</math>; instead using <math>B_1^+</math> the formula avoids subtraction:
where ''B''<sub>''n''&nbsp;+&nbsp;1</sub>(''m'') is the (''n''&nbsp;+&nbsp;1)th-degree ].
:<math display=inline> \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}+\binom{m+1} 1 B^+_1 n^m+\binom{m+1} 2B_2n^{m-1}+\cdots+\binom{m+1}mB_mn\right). </math>


=== Reconstruction of "Summae Potestatum" ===
Bernoulli numbers may be calculated by using the following ] formula:
[[File:JakobBernoulliSummaePotestatum.png|thumb|right|upright=1.5|Jakob Bernoulli's "Summae Potestatum", 1713{{efn|Translation of the text:
" ... And if to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list:<br>
Sums of powers<br>
<math>\textstyle \int n = \sum_{k=1}^n k = \frac {1}{2} n^2 + \frac {1}{2} n </math><br>
::::⋮
<math>\textstyle \int n^{10} = \sum_{k=1}^n k^{10} = \frac {1}{11} n^{11} + \frac {1}{2} n^{10} + \frac {5}{6} n^9 - 1 n^7 + 1 n^5 - \frac {1}{2} n^3 + \frac {5}{66} n </math><br>
Indeed one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these circuitous computations: For <math>\textstyle c </math> is taken as the exponent of any power, the sum of all <math>\textstyle n^c </math> is produced or<br>
<math>\textstyle \int n^c = \sum_{k=1}^n k^c = \frac {1}{c+1} n^{c+1} + \frac {1}{2} n^c + \frac {c}{2} An^{c-1} + \frac {c(c-1)(c-2)}{2\cdot 3\cdot4} Bn^{c-3} + \frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4 \cdot 5 \cdot 6} Cn^{c-5} + \frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} Dn^{c-7} + \cdots </math><br>
and so forth, the exponent of its power <math>n</math> continually diminishing by 2 until it arrives at <math>n</math> or <math>n^2</math>. The capital letters <math>\textstyle A, B, C, D, </math> etc. denote in order the coefficients of the last terms for <math>\textstyle \int n^2 , \int n^4 , \int n^6 , \int n^8 </math>, etc. namely<br>
<math>\textstyle A = \frac {1}{6} , B = - \frac {1}{30} , C = \frac {1}{42} , D = - \frac {1}{30} </math>."<br>
* {{citation |last=Smith |first=David Eugene |date= 1929 |chapter=Jacques (I) Bernoulli: On the 'Bernoulli Numbers' |title= A Source Book in Mathematics |chapter-url=https://archive.org/details/sourcebookinmath00smit/page/85 |location=New York |publisher=McGraw-Hill Book Co. |pages=85–90 }}
* {{citation |last=Bernoulli |first=Jacob |date=1713 |title=Ars Conjectandi |url=https://archive.org/details/jacobibernoulli00bern/page/97 |location=Basel |publisher=Impensis Thurnisiorum, Fratrum |pages=97–98 |language=la |doi=10.5479/sil.262971.39088000323931}}
}}]]


The Bernoulli numbers {{OEIS2C|id=A164555}}(n)/{{OEIS2C|id=A027642}}(n) were introduced by Jakob Bernoulli in the book '']'' published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted {{math|''A''}}, {{math|''B''}}, {{math|''C''}} and {{math|''D''}} by Bernoulli are mapped to the notation which is now prevalent as {{math|''A'' {{=}} ''B''<sub>2</sub>}}, {{math|''B'' {{=}} ''B''<sub>4</sub>}}, {{math|''C'' {{=}} ''B''<sub>6</sub>}}, {{math|''D'' {{=}} ''B''<sub>8</sub>}}. The expression {{math|''c''·''c''−1·''c''−2·''c''−3}} means {{math|''c''·(''c''−1)·(''c''−2)·(''c''−3)}} – the small dots are used as grouping symbols. Using today's terminology these expressions are ] {{math|''c''<sup>{{underline|''k''}}</sup>}}. The factorial notation {{math|''k''!}} as a shortcut for {{math|1 × 2 × ... × ''k''}} was not introduced until 100 years later. The integral symbol on the left hand side goes back to ] in 1675 who used it as a long letter {{math|''S''}} for "summa" (sum).{{efn|The {{harvp|''Mathematics Genealogy Project''|n.d.}} shows Leibniz as the academic<!--not doctoral--> advisor of Jakob Bernoulli. See also {{harvp|Miller|2017}}.}} The letter {{math|''n''}} on the left hand side is not an index of ] but gives the upper limit of the range of summation which is to be understood as {{math|1, 2, ..., ''n''}}. Putting things together, for positive {{math|''c''}}, today a mathematician is likely to write Bernoulli's formula as:
:<math>\sum_{j=0}^m{m+1\choose{j}}B_j = 0</math>


: <math> \sum_{k=1}^n k^c = \frac{n^{c+1}}{c+1}+\frac 1 2 n^c+\sum_{k=2}^c \frac{B_k}{k!} c^{\underline{k-1}}n^{c-k+1}.</math>
for ''m'' > 0, and ''B''<sub>0</sub> = 1.


This formula suggests setting {{math|''B''<sub>1</sub> {{=}} {{sfrac|1|2}}}} when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the ] {{math|''c''<sup>{{underline|''k''−1}}</sup>}} has for {{math|''k'' {{=}} 0}} the value {{math|{{sfrac|1|''c'' + 1}}}}.{{sfnp|Graham|Knuth|Patashnik|1989|loc=Section 2.51}} Thus Bernoulli's formula can be written
An alternative convention for the Bernoulli numbers is to set ''B''<sub>1</sub> = 1/2 rather than −1/2. If this convention is used, then all the Bernoulli numbers may be calculated by a different ] formula without further qualification:
: <math> \sum_{k=1}^n k^c = \sum_{k=0}^c \frac{B_k}{k!}c^{\underline{k-1}} n^{c-k+1}</math>


if {{math|''B''<sub>1</sub> {{=}} 1/2}}, recapturing the value Bernoulli gave to the coefficient at that position.
:<math>\sum_{j=0}^m{m+1\choose{j}}B_j = m+1.</math>


The formula for <math>\textstyle \sum_{k=1}^n k^9</math> in the first half of the quotation by Bernoulli above contains an error at the last term; it should be <math>-\tfrac {3}{20}n^2</math> instead of <math>-\tfrac {1}{12}n^2</math>.
Dividing both sides by ''m'' + 1 then gives a form suggestive of the connection with the Riemann zeta function if the ''j''=0 case is understood as a limit to deal with the pole at ζ(1):


== Definitions ==
:<math>\sum_{j=0}^m{m\choose{j-1}}\frac{B_j}{j} = 1</math>
Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned:


* a recursive equation,
The terms of this sum are the coefficients given by ] for the closed form of <math>\sum_{x=1}^n x^m</math> and so this recursive definition is merely reflecting the fact that these sums evaluate to 1 when ''n''=1 for any ''m''. In the alternative convention, the generating function is
* an explicit formula,
* a generating function,
* an integral expression.


For the proof of the ] of the four approaches.<ref>See {{harvp|Ireland|Rosen|1990}} or {{harvp|Conway|Guy|1996}}.</ref>
:<math>\frac{xe^x}{e^x-1}.</math>


=== Recursive definition ===
== Values of the Bernoulli numbers ==
The Bernoulli numbers obey the sum formulas{{r|Weisstein2016}}
: <math> \begin{align} \sum_{k=0}^{m}\binom {m+1} k B^{-{}}_k &= \delta_{m, 0} \\ \sum_{k=0}^{m}\binom {m+1} k B^{+{}}_k &= m+1 \end{align}</math>
where <math>m=0,1,2...</math> and {{math|''δ''}} denotes the ].


The first of these is sometimes written<ref>Jordan (1950) p 233</ref> as the formula (for m > 1)
''B''<sub>''n''</sub> = 0 for all odd ''n'' other than 1. ''B''<sub>1</sub> = 1/2 or −1/2 depending on the convention adopted (see below). The first few non-zero Bernoulli numbers (sequences {{OEIS2C|A027641}} and {{OEIS2C|A027642}} in ]) and some larger ones are listed below.
<math display=block>(B+1)^m-B_m=0,</math>
where the power is expanded formally using the binomial theorem and <math>B^k</math> is replaced by <math>B_k</math>.


Solving for <math>B^{\mp{}}_m</math> gives the recursive formulas<ref>Ireland and Rosen (1990) p 229</ref>
<center> <table cellpadding="3" border="1">
: <math>\begin{align}
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''n''</td>
B_m^{-{}} &= \delta_{m, 0} - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^{-{}}_k}{m - k + 1} \\
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''N''</td>
B_m^+ &= 1 - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^+_k}{m - k + 1}.
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''D''</td>
\end{align}</math>
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''B''<sub>''n''</sub> = ''N'' / ''D''</td>
<tr>
<td align="right"><tt>0 </tt></td>
<td align="right"><tt>1 </tt></td>
<td align="right"><tt>1 </tt></td>
<td align="right"><tt>+1.00000000000 </tt></td>
</tr>
<tr>
<td align="right"><tt>1 </tt></td>
<td align="right"><tt>-1 </tt></td>
<td align="right"><tt>2 </tt></td>
<td align="right"><tt>-0.50000000000 </tt></td>
</tr>
<tr>
<td align="right"><tt>2 </tt></td>
<td align="right"><tt>1 </tt></td>
<td align="right"><tt>6 </tt></td>
<td align="right"><tt>+0.16666666667 </tt></td>
</tr>
<tr>
<td align="right"><tt>4 </tt></td>
<td align="right"><tt>-1 </tt></td>
<td align="right"><tt>30 </tt></td>
<td align="right"><tt>-0.03333333333 </tt></td>
</tr>
<tr>
<td align="right"><tt>6 </tt></td>
<td align="right"><tt>1 </tt></td>
<td align="right"><tt>42 </tt></td>
<td align="right"><tt>+0.02380952381 </tt></td>
</tr>
<tr>
<td align="right"><tt>8 </tt></td>
<td align="right"><tt>-1 </tt></td>
<td align="right"><tt>30 </tt></td>
<td align="right"><tt>-0.03333333333 </tt></td>
</tr>
<tr>
<td align="right"><tt>10 </tt></td>
<td align="right"><tt>5 </tt></td>
<td align="right"><tt>66 </tt></td>
<td align="right"><tt>+0.07575757576 </tt></td>
</tr>
<tr>
<td align="right"><tt>12 </tt></td>
<td align="right"><tt>-691 </tt></td>
<td align="right"><tt>2730 </tt></td>
<td align="right"><tt>-0.25311355311 </tt></td>
</tr>
<tr>
<td align="right"><tt>14 </tt></td>
<td align="right"><tt>7 </tt></td>
<td align="right"><tt>6 </tt></td>
<td align="right"><tt>+1.16666666667 </tt></td>
</tr>
<tr>
<td align="right"><tt>16 </tt></td>
<td align="right"><tt>-3617 </tt></td>
<td align="right"><tt>510 </tt></td>
<td align="right"><tt>-7.09215686275 </tt></td>
</tr>
<tr>
<td align="right"><tt>18 </tt></td>
<td align="right"><tt>43867 </tt></td>
<td align="right"><tt>798 </tt></td>
<td align="right"><tt>+54.9711779448 </tt></td>
</tr>
</table></center>


=== Explicit definition ===
== Efficient computation of Bernoulli numbers ==
In 1893 ] listed a total of 38 explicit formulas for the Bernoulli numbers,{{r|Saalschütz1893}} usually giving some reference in the older literature. One of them is (for <math>m\geq 1</math>):
:<math>\begin{align}
B^-_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j j^m \\
B^+_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j (j + 1)^m.
\end{align}</math>


=== Generating function ===
In some applications it is useful to be able to compute the Bernoulli numbers ''B''<sub>0</sub> through ''B''<sub>''p''&nbsp;−&nbsp;3</sub> modulo ''p'', where ''p'' is a prime; for example to test whether ] holds for ''p'', or even just to determine whether ''p'' is an ]. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) ''p''<sup>2</sup> arithmetic operations would be required. Fortunately, faster methods have been developed {{harv|Buhler|Crandall|Ernvall|Metsankyla|2001}} which require only O(''p'' (log ''p'')<sup>2</sup>) operations (see ]).
The exponential ]s are
:<math>\begin{alignat}{3}
\frac{t}{e^t - 1} &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} -1 \right) &&= \sum_{m=0}^\infty \frac{B^{-{}}_m t^m}{m!}\\
\frac{te^t}{e^t - 1} = \frac{t}{1 - e^{-t}} &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} +1 \right) &&= \sum_{m=0}^\infty \frac{B^+_m t^m}{m!}.
\end{alignat}</math>
where the substitution is <math>t \to - t</math>. The two generating functions only differ by ''t''.
{{collapse top|title=Proof}}
If we let <math>F(t)=\sum_{i=1}^\infty f_it^i</math> and <math>G(t)=1/(1+F(t))=\sum_{i=0}^\infty g_it^i</math> then


:<math>G(t)=1-F(t)G(t).</math>
David Harvey {{harv|Harvey|2008}} describes an algorithm for computing Bernoulli numbers by computing ''B''<sub>''n''</sub> modulo ''p'' for
many small primes ''p'', and then reconstructing ''B''<sub>''n''</sub> via the ]. Harvey writes that the ] ] of this algorithm is O(''n''<sup>2</sup> log(''n'')<sup>2+eps</sup>) and claims that this ] is significantly faster than implementations based on other methods. Harvey's implementation is included in ] since version 3.1. Using this implementation Harvey computed ''B''<sub>''n''</sub> for ''n''&nbsp;=&nbsp;10<sup>8</sup>, which is a new record (October 2008). Prior to that Bernd Kellner {{harv|Kellner|2002}}<!--A more specific citation would be preferrable.--> computed ''B''<sub>''n''</sub> to full precision for ''n''&nbsp;=&nbsp;10<sup>6</sup> on December 2002 and Oleksandr Pavlyk {{harv|Pavlyk|2008}} for ''n''&nbsp;=&nbsp;10<sup>7</sup> with 'Mathematica' on April 2008.


Then <math>g_0=1</math> and for <math>m>0</math> the m{{sup|th}} term in the series for <math>G(t)</math> is:
<center> <table cellpadding="3" border="1">
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''Computer''</td>
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''Year''</td>
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''n''</td>
<td style="line-height: 1.5em; background:#D9DECD; color:red; text-align:center">''Digits''</td>
<tr>
<td align="right"><tt>J. Bernoulli </tt></td>
<td align="center"><tt>~1689</tt></td>
<td align="left"><tt>10</tt></td>
<td align="left"><tt>1</tt></td>
</tr>
<tr>
<td align="right"><tt>L. Euler</tt></td>
<td align="center"><tt>1748</tt></td>
<td align="left"><tt>30</tt></td>
<td align="left"><tt>8</tt></td>
</tr>
<tr>
<td align="right"><tt>J.C. Adams</tt></td>
<td align="center"><tt>1878</tt></td>
<td align="left"><tt>62</tt></td>
<td align="left"><tt>36</tt></td>
</tr>
<tr>
<td align="right"><tt>D.E. Knuth, T.J. Buckholtz</tt></td>
<td align="center"><tt>1967</tt></td>
<td align="left"><tt>360</tt></td>
<td align="left"><tt>478</tt></td>
</tr>


:<math>g_mt^i=-\sum_{j=0}^{m-1}f_{m-j}g_jt^m</math>
<tr>
<td align="right"><tt>G. Fee, S. Plouffe</tt></td>
<td align="center"><tt>1996</tt></td>
<td align="left"><tt>10000</tt></td>
<td align="left"><tt>27677</tt></td>
</tr>
<tr>
<td align="right"><tt>G. Fee, S. Plouffe</tt></td>
<td align="center"><tt>1996</tt></td>
<td align="left"><tt>100000</tt></td>
<td align="left"><tt>376755</tt></td>
</tr>
<tr>
<td align="right"><tt>B.C. Kellner</tt></td>
<td align="center"><tt>2002</tt></td>
<td align="left"><tt>1000000</tt></td>
<td align="left"><tt>4767529</tt></td>
</tr>
<tr>
<td align="right"><tt>O. Pavlyk</tt></td>
<td align="center"><tt>2008</tt></td>
<td align="left"><tt>10000000</tt></td>
<td align="left"><tt>57675260</tt></td>
</tr>
<tr>
<td align="right"><tt>D. Harvey</tt></td>
<td align="center"><tt>2008</tt></td>
<td align="left"><tt>100000000</tt></td>
<td align="left"><tt>676752569</tt></td>
</tr>
<caption>History of the computation of Bernoulli numbers</caption>
</table>
</center>
''Digits'' is to be understood as the exponent of 10 when B(n) is written as a real in normalized ].


If
== Reconstruction of 'Summae Potestatum' ==


:<math>F(t)=\frac{e^t-1}t-1=\sum_{i=1}^\infty \frac{t^i}{(i+1)!}</math>
<center><table>
<tr style="padding: 1.5em; line-height: 1.5em; background:#D9DECD; color:red; text-align:center">
<td>Jakob Bernoulli's ''Summae Potestatum'', 1713</td></tr>
<tr><td>
]
</td></tr></table></center>


then we find that
The Bernoulli numbers were introduced by Jakob Bernoulli in the book ']' published posthumously in 1713. The main formula can be seen in the second half of the facsimile given above. The constant coefficients denoted ''A'', ''B'', ''C'' and ''D'' by Bernoulli are mapped to the notation which is now prevalent as ''A''&nbsp;=&nbsp;''B''<sub>2</sub>, ''B''&nbsp;=&nbsp;''B''<sub>4</sub>, ''C''&nbsp;=&nbsp;''B''<sub>6</sub>, ''D''&nbsp;=&nbsp;''B''<sub>8</sub>. In the expression ''c''·''c''−1·''c''−2·''c''−3 the small dots are used as grouping symbols, not as signs for multiplication. Using today's terminology these expressions are falling factorial powers <math>c^{\underline{k}}</math>. The factorial notation ''k''! as a shortcut for 1×2×...×''k'' was not introduced until 100 years later. The integral symbol on the left hand side goes back to ] in 1675 who used it as a long letter ''S'' for "summa" (sum). (The ''Mathematics Genealogy Project'' <ref group="~"> </ref>
shows Leibniz as the doctoral adviser of Jakob Bernoulli. See also the ''Earliest Uses of Symbols of Calculus'' <ref group="~"> </ref>.) The letter ''n'' on the left hand side is not an index of ] but gives the upper limit of the range of summation which is to be understood as 1,2,...,''n''. Putting things together, for positive c, today a mathematician is likely to write Bernoulli's formula as


:<math>G(t)=t/(e^t-1)</math>
: <math> \sum_{0 < k \leq n} k^{c} = \frac{n^{c+1}}{c+1}+\frac{1}{2}n^c+\sum_{k \geq 2}\frac{B_{k}}{k!}c^{\underline{k-1}}n^{c-k+1}\ .</math>


:<math>\begin{align}
In fact this formula imperatively suggests to set ''B''<sub>1</sub> = 1/2 when switching from the so
m!g_m&=-\sum_{j=0}^{m-1}\frac{m!}{j!}\frac{j!g_j}{(m-j+1)!}\\
called 'archaic' enumeration which uses only the even indices 2,4,.. to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the falling factorial <math>c^{\underline{k-1}}</math> has for ''k''&nbsp;=&nbsp;0 the value 1/(''c''&nbsp;+&nbsp;1). (See the explanation in <ref group="~">''Concrete Mathematics'', (2.51)</ref>.) Thus Bernoulli's formula can and has to be written
&=-\frac 1{m+1}\sum_{j=0}^{m-1}\binom{m+1}jj!g_j\\
\end{align}</math>


showing that the values of <math>i!g_i</math> obey the recursive formula for the Bernoulli numbers <math>B^-_i</math>.
: <math> \sum_{0 < k \leq n} k^{c} = \sum_{k \geq 0}\frac{B_{k}}{k!}c^{\underline{k-1}}n^{c-k+1}
{{collapse bottom}}
\ .</math>


The (ordinary) generating function
if ''B''<sub>1</sub> stands for the value Bernoulli himself has given to the coefficient at that position.
: <math> z^{-1} \psi_1(z^{-1}) = \sum_{m=0}^{\infty} B^+_m z^m</math>


is an ]. It contains the ] {{math|''ψ''<sub>1</sub>}}.
== Different viewpoints and conventions ==


=== Integral Expression ===
The Bernoulli numbers can be regarded from four main viewpoints:
From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers:


:<math>B_{2n} = 4n (-1)^{n+1} \int_0^{\infty} \frac{t^{2n-1}}{e^{2 \pi t} -1 } \mathrm{d} t </math>
* as standalone arithmetical objects,
* as combinatorial objects,
* as values of a sequence of certain polynomials,
* as values of the Riemann zeta function.


== Bernoulli numbers and the Riemann zeta function ==
Each of these viewpoints leads to a set of more or less different conventions.


]
<ul>
<li>
''Bernoulli numbers as standalone arithmetical objects.''<br>
Associated sequence: 1/6, −1/30, 1/42, −1/30,... <br>
This is the viewpoint of Jakob Bernoulli. (See the cutout from his ''Ars Conjectandi'', first edition, 1713). The Bernoulli numbers are understood as numbers, recursive in nature, invented to solve a certain arithmetical problem, the summation of powers, which is the ''paradigmatic application'' of the Bernoulli numbers. It is misleading to call this viewpoint &#39;archaic&#39;. For example ] uses it in his highly acclaimed book ''A Course in Arithmetic'' which is a standard textbook used at many universities today.
</li>


The Bernoulli numbers can be expressed in terms of the ]:
<li>
''Bernoulli numbers as combinatorial objects.'' <br>
Associated sequence: 1,&nbsp;+1/2,&nbsp;1/6,&nbsp;0,.... <br>
This view focuses on the connection between Stirling numbers and Bernoulli numbers and arises naturally in the calculus of finite differences. In its most general and compact form this connection is summarized by the definition of the ''Stirling polynomials'' σ<sub>''n''</sub>(''x''), formula (6.52) in ''Concrete Mathematics'' by Graham, Knuth and Patashnik.


:{{math|1=''B''{{su|p=+|b=''n''}} = −''n ζ''(1 − ''n'')}} &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for {{math|''n'' ≥ 1}}&nbsp;.
: <math> \left(\frac{ze^{z}}{e^{z}-1} \right)^{x} = x\sum_{n\geq0}\sigma_{n}(x)z^{n}</math>


Here the argument of the zeta function is ''0 ''or negative. As <math>\zeta(k)</math> is zero for negative even integers (the ]), if ''n>1'' is odd, <math>\zeta(1-n)</math> is zero.
In consequence ''B''<sub>''n''</sub>&nbsp;=&nbsp;''n''! σ<sub>''n''</sub>(1) for ''n''&nbsp;≥&nbsp;0.
</li>


By means of the zeta ] and the gamma ] the following relation can be obtained:{{sfnp|Arfken|1970|p=279}}
<li>
''Bernoulli numbers as values of a sequence of certain polynomials.'' <br>
Assuming the Bernoulli polynomials as already introduced
the Bernoulli numbers can be defined in two different ways: <br>
''B''<sub>''n''</sub>&nbsp;=&nbsp;''B''<sub>''n''</sub>(0). Associated sequence: 1, −1/2, 1/6, 0,.... <br>
''B''<sub>''n''</sub>&nbsp;=&nbsp;''B''<sub>''n''</sub>(1). Associated sequence: 1, +1/2, 1/6, 0,....<br>
The two definitions differ only in the sign of ''B''<sub>1</sub>. The choice ''B''<sub>''n''</sub>&nbsp;=&nbsp;''B''<sub>''n''</sub>(0) is the convention used in the ''Handbook of Mathematical Functions''.
</li>


:<math> B_{2n} = \frac {(-1)^{n+1}2(2n)!} {(2\pi)^{2n}} \zeta(2n) \quad </math> for {{math|''n'' ≥ 1}}&nbsp;.
<li>
''Bernoulli numbers as values of the Riemann zeta function.'' <br>
Associated sequence: 1, +1/2, 1/6, 0,.... <br>


Now the argument of the zeta function is positive.
]
This convention agrees with the convention ''B''<sub>''n''</sub>&nbsp;=&nbsp;''B''<sub>''n''</sub>(1) (for example J. Neukirch and M. Kaneko). The sign &#39;+&#39; for ''B''<sub>''1''</sub> matches the representation of the Bernoulli numbers by the ]. In fact the identity ''n''ζ(1−''n'')&nbsp;=&nbsp;(−1)<sup>''n''+1</sup>''B''<sub>''n''</sub> valid for all ''n''&nbsp;&gt;&nbsp;0 is then replaced by the simpler ''n''ζ(1−''n'')&nbsp;=&nbsp;-''B''<sub>''n''</sub>. (See the paper of S. C. Woon.)<br>
(Note that in the foregoing equation for ''n''&nbsp;=&nbsp;0 and ''n''&nbsp;=&nbsp;1 the expression −''n''ζ(1&nbsp;−&nbsp;''n'') is to be understood as lim<sub>''x''&nbsp;→&nbsp;''n''</sub>&nbsp;−''x''ζ(1&nbsp;−&nbsp;''x'').)
</li>
</ul>


It then follows from {{math|''&zeta;'' &rarr; 1}} ({{math|''n'' &rarr; &infin;}}) and ] that
<center>
: <math> |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n} \quad </math> for {{math|''n'' &rarr; &infin;}}&nbsp;.
{| border="5" cellspacing="5" cellpadding="2"

| <math> \ B_n = n!\sigma_{n}(1) = B_n(1) = -n\zeta(1-n) \quad (n \geq 0) \ </math>
== Efficient computation of Bernoulli numbers ==

In some applications it is useful to be able to compute the Bernoulli numbers {{math|''B''<sub>0</sub>}} through {{math|''B''<sub>''p'' − 3</sub>}} modulo {{mvar|p}}, where {{mvar|p}} is a prime; for example to test whether ] holds for {{mvar|p}}, or even just to determine whether {{mvar|p}} is an ]. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) {{math|''p''<sup>2</sup>}} arithmetic operations would be required. Fortunately, faster methods have been developed{{r|BuhlerCraErnMetShokrollahi2001}} which require only {{math|''O''(''p'' (log ''p'')<sup>2</sup>)}} operations (see ]).

David Harvey{{r|Harvey2010}} describes an algorithm for computing Bernoulli numbers by computing {{math|''B''<sub>''n''</sub>}} modulo {{mvar|p}} for many small primes {{mvar|p}}, and then reconstructing {{math|''B''<sub>''n''</sub>}} via the ]. Harvey writes that the ] ] of this algorithm is {{math|''O''(''n''<sup>2</sup> log(''n'')<sup>2 + ''ε''</sup>)}} and claims that this ] is significantly faster than implementations based on other methods. Using this implementation Harvey computed {{math|''B''<sub>''n''</sub>}} for {{math|''n'' {{=}} 10<sup>8</sup>}}. Harvey's implementation has been included in ] since version 3.1. Prior to that, Bernd Kellner{{r|Kellner2002}}<!--A more specific citation would be preferable.--> computed {{math|''B''<sub>''n''</sub>}} to full precision for {{math|''n'' {{=}} 10<sup>6</sup>}} in December 2002 and Oleksandr Pavlyk{{r|Pavlyk29Apr2008}} for {{math|''n'' {{=}} 10<sup>7</sup>}} with ] in April 2008.

{{table alignment}}
:{| class="wikitable defaultright col1left"
! Computer !! Year !! ''n'' !! Digits*
|-
| J. Bernoulli || ~1689 || 10 || 1
|- |-
| L. Euler || 1748 || 30 || 8
|-
| J. C. Adams || 1878 || 62 || 36
|-
| D. E. Knuth, T. J. Buckholtz || 1967 || {{val|1672|fmt=gaps}} || {{val|3330|fmt=gaps}}
|-
| G. Fee, ] || 1996 || {{val|10000}} || {{val|27677}}
|-
| G. Fee, S. Plouffe || 1996 || {{val|100000}} || {{val|376755}}
|-
| B. C. Kellner || 2002 || {{val|1000000}} || {{val|4767529}}
|-
| O. Pavlyk || 2008 || {{val|10000000}} || {{val|57675260}}
|-
| D. Harvey || 2008 || {{val|100000000}} || {{val|676752569}}
|} |}
::<nowiki>*</nowiki> ''Digits'' is to be understood as the exponent of 10 when {{math|''B''<sub>''n''</sub>}} is written as a real number in normalized ].
</center>


== Application of the Bernoulli number == == Applications of the Bernoulli numbers ==
Arguably the most important application of the Bernoulli number in mathematics is their use in the ]. Assuming that ''f'' is a sufficiently often differentiable function the Euler-MacLaurin formula can be written as <ref group="~">''Concrete Mathematics'', (9.67).</ref>


=== Asymptotic analysis ===
: <math> \sum\limits_{a\leq k<b}f(k)=\int_{a}^{b}f(x)dx \ + \sum\limits_{k=1}^{m}\frac{B_{k}}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R(f,m) \ . </math>
Arguably the most important application of the Bernoulli numbers in mathematics is their use in the ]. Assuming that {{mvar|f}} is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as{{sfnp|Graham|Knuth|Patashnik|1989|loc=9.67}}


: <math>\sum_{k=a}^{b-1} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^-_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_-(f,m).</math>
This formulation assumes the convention ''B''<sub>1</sub> = -1/2. However, if one sets ''B''<sub>1</sub> = 1/2 then this formula can also be written as


This formulation assumes the convention {{math|1=''B''{{su|p=−|b=1}} = −{{sfrac|1|2}}}}. Using the convention {{math|1=''B''{{su|p=+|b=1}} = +{{sfrac|1|2}}}} the formula becomes
: <math> \sum\limits_{a<k\leq b}f(k)=\int_{a}^{b}f(x)dx\ \ +\ \sum\limits_{k=1}^{m}\frac{B_{k}}{k!} \left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R(f,m) \ . </math>


: <math>\sum_{k=a+1}^{b} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^+_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_+(f,m).</math>
Here ''f''<sup>(0)</sup> = ''f'' which is a commonly used notation identifying the zero-th derivative of ''f'' with ''f''. Moreover, let ''f''<sup>(-1)</sup> denote an ] of ''f''. By the ] <math>\int_{a}^{b}f(x)dx\ = f^{(-1)}(b) - f^{(-1)}(a)</math>. Thus the last formula can be further simplified to the following succinct form of the Euler-Maclaurin formula


Here <math>f^{(0)}=f</math> (i.e. the zeroth-order derivative of <math>f</math> is just <math>f</math>). Moreover, let <math>f^{(-1)}</math> denote an ] of <math>f</math>. By the ],
: <math> \sum\limits_{a<k\leq b}f(k)= \sum\limits_{k=0}^{m}\frac{B_{k}}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R(f,m) \ . </math>


: <math>\int_a^b f(x)\,dx = f^{(-1)}(b) - f^{(-1)}(a).</math>
This form is for example the source for the important Euler-MacLaurin expansion of the zeta function

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

: <math> \sum_{k=a+1}^{b} f(k)= \sum_{k=0}^m \frac{B_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m). </math>

This form is for example the source for the important Euler–Maclaurin expansion of the zeta function


: <math> \begin{align} : <math> \begin{align}
\zeta(s)& =\sum_{k=0}^{m}\frac{{B}_{k}}{k!} s^{\overline{k-1}} + R(s,m) \\ \zeta(s) & =\sum_{k=0}^m \frac{B^+_k}{k!} s^{\overline{k-1}} + R(s,m) \\
& =\frac{{B}_{0}}{0!}s^{\overline{-1}} +\frac{{B}_{1}}{1!} s^{\overline{0}} +\frac{{B}_{2}}{2!} s^{\overline{1}} +\cdots+R(s,m) \\ & = \frac{B_0}{0!}s^{\overline{-1}} + \frac{B^+_1}{1!} s^{\overline{0}} + \frac{B_2}{2!} s^{\overline{1}} +\cdots+R(s,m) \\
& =\frac{1}{s-1}+\frac{1}{2}+\frac{1}{12}s +\cdots+R(s,m) \ . \\ & = \frac{1}{s-1} + \frac{1}{2} + \frac{1}{12}s + \cdots + R(s,m).
\end{align} </math> \end{align} </math>


Here <math>s^{\overline{k}}</math> denotes the rising factorial power.<ref group="~">''Concrete Mathematics'', (2.44) and (2.52)</ref> Here {{math|''s''<sup>{{overline|''k''}}</sup>}} denotes the ].{{sfnp|Graham|Knuth|Patashnik|1989|loc=2.44, 2.52}}


Bernoulli numbers are also frequently used in other kinds of ]s. The following example is the classical Poincaré-type asymptotic expansion of the ] {{math|''ψ''}}.
==Combinatorial definitions==


:<math>\psi(z) \sim \ln z - \sum_{k=1}^\infty \frac{B^+_k}{k z^k} </math>
The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the ].


=== Connection with the Worpitzky number === === Sum of powers ===
{{main|Faulhaber's formula}}
Bernoulli numbers feature prominently in the ] expression of the sum of the {{math|''m''}}th powers of the first {{math|''n''}} positive integers. For {{math|''m'', ''n'' ≥ 0}} define


:<math>S_m(n) = \sum_{k=1}^n k^m = 1^m + 2^m + \cdots + n^m. </math>
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function ''n''! and the power function ''k''<sup>''m''</sup> is employed. The signless Worpitzky numbers are defined as


This expression can always be rewritten as a ] in {{math|''n''}} of degree {{math|''m'' + 1}}. The ]s of these polynomials are related to the Bernoulli numbers by '''Bernoulli's formula''':
: <math> W_{n,k}=\sum_{v=0}^{k}(-1)^{v+k} \left(v+1\right)^{n} \frac{k!}{v!(k-v)!} \ . </math>
: <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m \binom{m + 1}{k} B^+_k n^{m + 1 - k} = m! \sum_{k=0}^m \frac{B^+_k n^{m + 1 - k}}{k! (m+1-k)!} ,</math>
where {{math|<big><big>(</big></big>{{su|p=''m'' + 1|b=''k''|a=c}}<big><big>)</big></big>}} denotes the ].


For example, taking {{math|''m''}} to be 1 gives the ]s {{math|0, 1, 3, 6, ...}} {{OEIS2C|id=A000217}}.
They can also be expressed through the ]


: <math> W_{n,k}=k! \left\{\begin{matrix} n+1 \\ k+1 \end{matrix}\right\}.</math> :<math> 1 + 2 + \cdots + n = \frac{1}{2} (B_0 n^2 + 2 B^+_1 n^1) = \tfrac12 (n^2 + n).</math>


Taking {{math|''m''}} to be 2 gives the ]s {{math|0, 1, 5, 14, ...}} {{OEIS2C|id=A000330}}.
A Bernoulli number is then introduced as an inclusion-exclusion sum of Worpitzky numbers weighted by the sequence 1,&nbsp;1/2,&nbsp;1/3,...


: <math> B_{n}=\sum_{k=0}^{n}(-1)^{k}\frac{W_{n,k}}{k+1} \ . </math> : <math>1^2 + 2^2 + \cdots + n^2 = \frac{1}{3} (B_0 n^3 + 3 B^+_1 n^2 + 3 B_2 n^1) = \tfrac13 \left(n^3 + \tfrac32 n^2 + \tfrac12 n\right).</math>


Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
<center><table style="border: 3px solid #D9DECD;" >
: <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m (-1)^k \binom{m + 1}{k} B^{-{}}_k n^{m + 1 - k}.</math>
<tr style="padding: 1.5em; line-height: 1.5em; background:#D9DECD; color:red; text-align:center">
<td colspan="3" >Worpitzky's representation of the Bernoulli number</td></tr>
<tr>
<td>''B''<sub>0</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1</td>
</tr>
<tr>
<td>''B''<sub>1</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1&nbsp;−&nbsp;1/2</td>
</tr>
<tr>
<td>''B''<sub>2</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1&nbsp;−&nbsp;3/2&nbsp;+&nbsp;2/3</td>
</tr>
<tr>
<td>''B''<sub>3</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1&nbsp;−&nbsp;7/2&nbsp;+&nbsp;12/3&nbsp;−&nbsp;6/4</td>
</tr>
<tr>
<td>''B''<sub>4</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1&nbsp;−&nbsp;15/2&nbsp;+&nbsp;50/3&nbsp;−&nbsp;60/4&nbsp;+&nbsp;24/5</td>
</tr>
<tr>
<td>''B''<sub>5</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1&nbsp;−&nbsp;31/2&nbsp;+&nbsp;180/3&nbsp;−&nbsp;390/4&nbsp;+&nbsp;360/5&nbsp;−&nbsp;120/6</td>
</tr>
<tr>
<td>''B''<sub>6</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1/1&nbsp;−&nbsp;63/2&nbsp;+&nbsp;602/3&nbsp;−&nbsp;2100/4&nbsp;+&nbsp;3360/5&nbsp;−&nbsp;2520/6&nbsp;+&nbsp;720/7</td>
</tr>
</table></center>


Bernoulli's formula is sometimes called ] after ] who also found remarkable ways to calculate ].
This representation has ''B''<sub>1</sub>&nbsp;=&nbsp;1/2.


Faulhaber's formula was generalized by V. Guo and J. Zeng to a ].{{r|GuoZeng2005}}
=== Connection with the Stirling set number ===


===Taylor series===
A similar combinatorial representation derives from
The Bernoulli numbers appear in the ] expansion of many ] and ]s.


<math display="block">\begin{align}
: <math> B_{n}=\sum_{k=0}^{n}(-1)^{k+n}\frac{k!}{k+1}
\tan x &= \hphantom{{1\over x}} \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} }{(2n)!}\; x^{2n-1}, && \left|x \right| < \frac \pi 2. \\
\left\{\begin{matrix} n \\ k \end{matrix}\right\} \ . </math>
\cot x &= {1\over x} \sum_{n=0}^\infty \frac{(-1)^n B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi. \\
\tanh x &= \hphantom{{1\over x}} \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\;x^{2n-1}, && |x| < \frac \pi 2. \\
\coth x &= {1\over x} \sum_{n=0}^\infty \frac{B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi.
\end{align}</math>


===Laurent series===
Here the Bernoulli numbers are an inclusion-exclusion over the set of length-''n'' words, where the sum is taken over all words of length ''n'' with ''k'' distinct letters, and normalized by ''k''&nbsp;+&nbsp;1. The combinatorics of this representation can be seen from:
The Bernoulli numbers appear in the following ]:{{sfnp|Arfken|1970|p=463}}


: <math> B_{0}= +\left\vert \ \left\{ \varnothing \right\} \ \right\vert </math> ]: <math> \psi(z)= \ln z- \sum_{k=1}^\infty \frac {B_k^{+{}}} {k z^k} </math>
: <math> B_{1}= -\frac{1}{1} \left\vert \ \varnothing \ \right\vert + \frac{1}{2}\left\vert \left\{a\right\}\right\vert </math>


=== Use in topology ===
: <math> B_{2}= +\frac{1}{1} \left\vert \ \varnothing \ \right\vert - \frac{1}{2}\left\vert \left\{aa\right\}\right\vert +\frac{1}{3}\left\vert \left\{ab,ba\right\} \right\vert </math>
The ] for the order of the cyclic group of diffeomorphism classes of ] which bound ]s involves Bernoulli numbers. Let {{math|''ES''<sub>''n''</sub>}} be the number of such exotic spheres for {{Math|''n'' ≥ 2}}, then


:<math>\textit{ES}_n = (2^{2n-2}-2^{4n-3}) \operatorname{Numerator}\left(\frac{B_{4n}}{4n} \right) .</math>
: <math> B_{3}= -\frac{1}{1} \left\vert \ \varnothing \ \right\vert +\frac{1}{2}\left\vert \left\{aaa\right\}\right\vert -\frac{1}{3}\left\vert \left\{aab,aba,baa,abb,bab,bba\right\}\right\vert +\frac{1}{4}\left\vert \left\{abc,acb,bac,bca,cab,cba\right\} \right\vert </math>


The ] for the ] of a ] ] ] of ] 4''n'' also involves Bernoulli numbers.
=== Connection with the Stirling cycle number ===


== Connections with combinatorial numbers ==
Let <math>\left</math> denote the signless ]. The two main formulas relating these number to the Bernoulli number (''B''<sub>1</sub>&nbsp;=&nbsp;1/2) are
The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the ].


=== Connection with Worpitzky numbers ===
: <math> \frac{1}{m!}\sum_{k=0}^{m}(-1)^{k} \left B_{k}=\frac{1}{m+1}\ , </math>
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function {{math|''n''!}} and the power function {{math|''k<sup>m</sup>''}} is employed. The signless Worpitzky numbers are defined as


: <math> W_{n,k}=\sum_{v=0}^k (-1)^{v+k} (v+1)^n \frac{k!}{v!(k-v)!} . </math>
and the inversion of this sum (for ''n''&nbsp;≥&nbsp;0, ''m''&nbsp;≥&nbsp;0)


They can also be expressed through the ]
: <math> \frac{1}{m!}\sum_{k=0}^{m}(-1)^{k} \left B_{n+k} = A_{n,m} \ . </math>


: <math> W_{n,k}=k! \left\{ {n+1\atop k+1} \right\}.</math>
Here the number ''A''<sub>''n'',''m''</sub> are the rational Akiyama-Tanigawa number, the first few of which are displayed in the following table.


A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the ] 1,&nbsp;{{sfrac|1|2}},&nbsp;{{sfrac|1|3}},&nbsp;...
<center><table style="border: 3px solid #D9DECD;" >
<tr style="padding: 1.5em; line-height: 1.5em; background:#D9DECD; color:red; text-align:center">
<td colspan="6">Akiyama-Tanigawa number</td></tr>
<tr>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">''n \ m''</td>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">0</td>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">1</td>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">2</td>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">3</td>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">4</td>
</tr><tr>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">0</td>
<td align="center">1</td><td align="center">1/2</td><td align="center">1/3</td><td align="center">1/4</td><td align="center">1/5</td>
</tr><tr>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">1</td>
<td align="center">1/2</td><td align="center">1/3</td><td align="center">1/4</td><td align="center">1/5</td><td align="center">...</td>
</tr><tr>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">2</td>
<td align="center">1/6</td><td align="center">1/6</td><td align="center">3/20</td><td align="center">...</td><td align="center">...</td>
</tr><tr>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">3</td>
<td align="center">0</td><td align="center">1/30</td><td align="center">...</td><td align="center">...</td><td align="center">...</td>
</tr><tr>
<td style="background: rgb(217, 222, 205); line-height: 1.5em; text-align: center;">4</td>
<td align="center">-1/30</td><td align="center">...</td><td align="center">...</td><td align="center">...</td><td align="center">...</td>
</tr></table></center>


: <math> B_{n}=\sum_{k=0}^n (-1)^k \frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^n \frac{1}{k+1} \sum_{v=0}^k (-1)^v (v+1)^n {k \choose v}\ . </math>
These relations lead to a simple algorithm to compute the Bernoulli number. The input is the first row,&nbsp;''A''<sub>0,''m''</sub>&nbsp;=&nbsp;1/(''m''&nbsp;+&nbsp;1) and the output are the Bernoulli number in the first column ''A''<sub>''n'',0</sub>&nbsp;= ''B''<sub>''n''</sub> . This transformation is shown in pseudo-code below.


:{{math|1=''B''<sub>0</sub> = 1}}
<table align="center" style="border: 3px solid rgb(217, 222, 205);">
:{{math|1=''B''<sub>1</sub> = 1 − {{sfrac|1|2}}}}
<tr style="padding: 1.5em; background: rgb(217, 222, 205); line-height: 1.5em; color: red; text-align: center;">
:{{math|1=''B''<sub>2</sub> = 1 − {{sfrac|3|2}} + {{sfrac|2|3}}}}
<td >Akiyama-Tanigawa algorithm for ''B''<sub>''n''</sub></td></tr>
:{{math|1=''B''<sub>3</sub> = 1 − {{sfrac|7|2}} + {{sfrac|12|3}} − {{sfrac|6|4}}}}
<tr><td>Enter integer ''n''.<br />
:{{math|1=''B''<sub>4</sub> = 1 − {{sfrac|15|2}} + {{sfrac|50|3}} − {{sfrac|60|4}} + {{sfrac|24|5}}}}
For ''m'' from 0 by 1 to ''n'' do
:{{math|1=''B''<sub>5</sub> = 1 − {{sfrac|31|2}} + {{sfrac|180|3}} − {{sfrac|390|4}} + {{sfrac|360|5}} − {{sfrac|120|6}}}}
<table><tr>
:{{math|1=''B''<sub>6</sub> = 1 − {{sfrac|63|2}} + {{sfrac|602|3}} − {{sfrac|2100|4}} + {{sfrac|3360|5}} − {{sfrac|2520|6}} + {{sfrac|720|7}}}}
<td>&nbsp;&nbsp;&nbsp;</td>
<td >''A''&nbsp;←&nbsp;1/(''m''&nbsp;+&nbsp;1)</td>
</tr><tr>
<td>&nbsp;&nbsp;&nbsp;</td>
<td >For ''j'' from ''m'' by −1 to 1 do</td>
</tr><tr>
<td>&nbsp;&nbsp;&nbsp;</td>
<td ><table><tr>
<td>&nbsp;&nbsp;&nbsp;</td>
<td>''A''&nbsp;←&nbsp;j&nbsp;×&nbsp;(''A''&nbsp;−&nbsp;''A'')</td></tr>
</table></td></tr></table>
Return ''A''&nbsp; (which is ''B''<sub>''n''</sub>).</td></tr>
</table></center>


This representation has {{math|''B''{{su|p=+|b=1}} {{=}} +{{sfrac|1|2}}}}.
=== Connection with the Eulerian number ===


Consider the sequence {{math|''s<sub>n</sub>''}}, {{math|''n'' ≥ 0}}. From Worpitzky's numbers {{OEIS2C|id=A028246}}, {{OEIS2C|id=A163626}} applied to {{math|''s''<sub>0</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ...}} is identical to the Akiyama–Tanigawa transform applied to {{math|''s<sub>n</sub>''}} (see ]). This can be seen via the table:
The ] are the number of permutations of {1,2,...,''n''} with ''m'' ascents. They were introduced by Leonhard Euler in 1755 and in the notation of D. E. Knuth <ref group="~">''Concrete Mathematics'', Table 254 Euler's triangle.</ref> written as <math> \left \langle {n\atop m} \right \rangle .</math> The two main formulas connecting the Eulerian number to the Bernoulli number are:


:{| style="text-align:center"
:<math>\sum_{m=0}^{n}(-1)^{m} {\left \langle {n\atop m} \right \rangle} = 2^{n+1}(2^{n+1}-1) \frac{B_{n+1}}{n+1} \ ,</math>
|+ '''Identity of<br/>Worpitzky's representation and Akiyama–Tanigawa transform'''
|-
|1|| || || || || ||0||1|| || || || ||0||0||1|| || || ||0||0||0||1|| || ||0||0||0||0||1||
|-
|1||−1|| || || || ||0||2||−2|| || || ||0||0||3||−3|| || ||0||0||0||4||−4|| || || || || || ||
|-
|1||−3||2|| || || ||0||4||−10||6|| || ||0||0||9||−21||12|| || || || || || || || || || || || ||
|-
|1||−7||12||−6|| || ||0||8||−38||54||−24|| || || || || || || || || || || || || || || || || || ||
|-
|1||−15||50||−60||24|| || || || || || || || || || || || || || || || || || || || || || || || ||
|-
|}


The first row represents {{math|''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ''s''<sub>4</sub>}}.
:<math>\sum_{m=0}^{n}(-1)^{m} {\left \langle {n\atop m} \right \rangle} {\binom{n}{m}}^{-1} = (n+1) B_{n} \ .</math>


Hence for the second fractional Euler numbers {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}):
Both formulas are valid for ''n''&nbsp;≥&nbsp;0 if ''B''<sub>1</sub> is set to 1/2. If ''B''<sub>1</sub> is set to −1/2 they are valid only for ''n''&nbsp;≥&nbsp;1 and ''n''&nbsp;≥&nbsp;2 respectively.


:{{math|1= ''E''<sub>0</sub> = 1}}
==Asymptotic approximation==
:{{math|1= ''E''<sub>1</sub> = 1 − {{sfrac|1|2}}}}
:{{math|1= ''E''<sub>2</sub> = 1 − {{sfrac|3|2}} + {{sfrac|2|4}}}}
:{{math|1= ''E''<sub>3</sub> = 1 − {{sfrac|7|2}} + {{sfrac|12|4}} − {{sfrac|6|8}}}}
:{{math|1= ''E''<sub>4</sub> = 1 − {{sfrac|15|2}} + {{sfrac|50|4}} − {{sfrac|60|8}} + {{sfrac|24|16}}}}
:{{math|1= ''E''<sub>5</sub> = 1 − {{sfrac|31|2}} + {{sfrac|180|4}} − {{sfrac|390|8}} + {{sfrac|360|16}} − {{sfrac|120|32}}}}
:{{math|1= ''E''<sub>6</sub> = 1 − {{sfrac|63|2}} + {{sfrac|602|4}} − {{sfrac|2100|8}} + {{sfrac|3360|16}} − {{sfrac|2520|32}} + {{sfrac|720|64}}}}


] expressed the Bernoulli numbers in terms of the ] as A second formula representing the Bernoulli numbers by the Worpitzky numbers is for {{math|''n'' ≥ 1}}


:<math>B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left. </math> : <math> B_n=\frac n {2^{n+1}-2}\sum_{k=0}^{n-1} (-2)^{-k}\, W_{n-1,k} . </math>


The simplified second Worpitzky's representation of the second Bernoulli numbers is:
It then follows from the ] that, as ''n'' goes to infinity,


{{OEIS2C|id=A164555}} ({{math|''n'' + 1}}) / {{OEIS2C|id=A027642}}({{math|''n'' + 1}}) = {{math|{{sfrac|''n'' + 1|2<sup>''n'' + 2</sup> − 2}}}} × {{OEIS2C|id=A198631}}({{math|''n''}}) / {{OEIS2C|id=A006519}}({{math|''n'' + 1}})
: <math> |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n}. </math>

which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:

:{{math|{{sfrac|1|2}}, {{sfrac|1|6}}, 0, −{{sfrac|1|30}}, 0, {{sfrac|1|42}}, ... {{=}} ({{sfrac|1|2}}, {{sfrac|1|3}}, {{sfrac|3|14}}, {{sfrac|2|15}}, {{sfrac|5|62}}, {{sfrac|1|21}}, ...) × (1, {{sfrac|1|2}}, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, ...)}}

The numerators of the first parentheses are {{OEIS2C|id=A111701}} (see ]).

=== Connection with Stirling numbers of the second kind ===

If one defines the ] {{math|''B<sub>k</sub>''(''j'')}} as:{{r|Rademacher1973}}

:<math> B_k(j)=k\sum_{m=0}^{k-1}\binom{j}{m+1}S(k-1,m)m!+B_k </math>

where {{math|''B<sub>k</sub>''}} for {{math|''k'' {{=}} 0, 1, 2,...}} are the Bernoulli numbers,
and {{math|''S(k,m)''}} is a ].

One also has the following for Bernoulli polynomials,{{r|Rademacher1973}}

:<math> B_k(j)=\sum_{n=0}^k \binom{k}{n} B_n j^{k-n}. </math>

The coefficient of {{mvar|j}} in {{math|<big><big>(</big></big>{{su|p=''j''|b=''m'' + 1|a=c}}<big><big>)</big></big>}} is {{math|{{sfrac|(−1)<sup>''m''</sup>|''m'' + 1}}}}.

Comparing the coefficient of {{mvar|j}} in the two expressions of Bernoulli polynomials, one has:

: <math> B_k=\sum_{m=0}^{k-1} (-1)^m \frac{m!}{m+1} S(k-1,m)</math>

(resulting in {{math|''B''<sub>1</sub> {{=}} +{{sfrac|1|2}}}}) which is an explicit formula for Bernoulli numbers and can be used to prove ].{{r|Boole1880|Gould1972|Apostol2010_197}}

=== Connection with Stirling numbers of the first kind ===
The two main formulas relating the unsigned ] {{math|<big><big></big></big>}} to the Bernoulli numbers (with {{math|''B''<sub>1</sub> {{=}} +{{sfrac|1|2}}}}) are

: <math> \frac{1}{m!}\sum_{k=0}^m (-1)^{k} \left B_k = \frac{1}{m+1}, </math>

and the inversion of this sum (for {{math|''n'' ≥ 0}}, {{math|''m'' ≥ 0}})

: <math> \frac{1}{m!}\sum_{k=0}^m (-1)^k \left B_{n+k} = A_{n,m}. </math>

Here the number {{math|''A''<sub>''n'',''m''</sub>}} are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.

:{| class="wikitable" style="text-align:center"
|+ Akiyama–Tanigawa number
! {{diagonal split header|{{mvar|n}}|{{mvar|m}}}}!!0!!1!!2!!3!!4
|-
! 0
| 1 || {{sfrac|1|2}} || {{sfrac|1|3}} || {{sfrac|1|4}} || {{sfrac|1|5}}
|-
! 1
| {{sfrac|1|2}} || {{sfrac|1|3}} || {{sfrac|1|4}} || {{sfrac|1|5}} || ...
|-
! 2
| {{sfrac|1|6}} || {{sfrac|1|6}} || {{sfrac|3|20}} || ... || ...
|-
! 3
| 0 || {{sfrac|1|30}} || ... || ... || ...
|-
! 4
| −{{sfrac|1|30}} || ... || ... || ... || ...
|}

The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See {{OEIS2C|id=A051714}}/{{OEIS2C|id=A051715}}.

An ''autosequence'' is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = {{OEIS2C|id=A000004}}, the autosequence is of the first kind. Example: {{OEIS2C|id=A000045}}, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: {{OEIS2C|id=A164555}}/{{OEIS2C|id=A027642}}, the second Bernoulli numbers (see {{OEIS2C|id=A190339}}). The Akiyama–Tanigawa transform applied to {{math|''2''<sup>−''n''</sup>}} = 1/{{OEIS2C|id=A000079}} leads to {{OEIS2C|id=A198631}} (''n'') / {{OEIS2C|id=A06519}} (''n'' + 1). Hence:

:{| class="wikitable" style="text-align:center"
|+ Akiyama–Tanigawa transform for the second Euler numbers
|-
! {{diagonal split header|{{mvar|n}}|{{mvar|m}}}} !! 0 !! 1 !! 2 !! 3 !! 4
|-
! 0
| 1 || {{sfrac|1|2}} || {{sfrac|1|4}} || {{sfrac|1|8}} || {{sfrac|1|16}}
|-
! 1
| {{sfrac|1|2}} || {{sfrac|1|2}} || {{sfrac|3|8}} || {{sfrac|1|4}} || ...
|-
! 2
| 0 || {{sfrac|1|4}} || {{sfrac|3|8}} || ... || ...
|-
! 3
| −{{sfrac|1|4}} || −{{sfrac|1|4}} || ... || ... || ...
|-
! 4
| 0 || ... || ... || ... || ...
|}

See {{OEIS2C|id=A209308}} and {{OEIS2C|id=A227577}}. {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}) are the second (fractional) Euler numbers and an autosequence of the second kind.

:({{sfrac|{{OEIS2C|id=A164555}} ({{math|''n'' + 2}})|{{OEIS2C|id=A027642}} ({{math|''n'' + 2}})}} = {{math|{{sfrac|1|6}}, 0, −{{sfrac|1|30}}, 0, {{sfrac|1|42}}, ...}}) × ( {{math|{{sfrac|2<sup>''n'' + 3</sup> − 2|''n'' + 2}}}} = {{math|3, {{sfrac|14|3}}, {{sfrac|15|2}}, {{sfrac|62|5}}, 21, ...}}) = {{sfrac|{{OEIS2C|id=A198631}} ({{math|''n'' + 1}})|{{OEIS2C|id=A006519}} ({{math|''n'' + 2}})}} = {{math|{{sfrac|1|2}}, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, ...}}.

Also valuable for {{OEIS2C|id=A027641}} / {{OEIS2C|id=A027642}} (see ]).

===Connection with Pascal's triangle===
There are formulas connecting Pascal's triangle to Bernoulli numbers{{efn|this formula was discovered (or perhaps rediscovered) by Giorgio Pietrocola. His demonstration is available in Italian language {{harv|Pietrocola|2008}}.}}
:<math> B^{+}_n=\frac{|A_n|}{(n+1)!}~~~</math>
where <math>|A_n|</math> is the determinant of a n-by-n ] part of ] whose elements are: <math>
a_{i, k} = \begin{cases}
0 & \text{if } k>1+i \\
{i+1 \choose k-1} & \text{otherwise}
\end{cases}
</math>

Example:

:<math> B^{+}_6 =\frac{\det\begin{pmatrix}
1& 2& 0& 0& 0& 0\\
1& 3& 3& 0& 0& 0\\
1& 4& 6& 4& 0& 0\\
1& 5& 10& 10& 5& 0\\
1& 6& 15& 20& 15& 6\\
1& 7& 21& 35& 35& 21
\end{pmatrix}}{7!}=\frac{120}{5040}=\frac 1 {42}
</math>

=== Connection with Eulerian numbers ===
There are formulas connecting ]s {{math|<big><big>⟨</big></big>{{su|p=''n''|b=''m''|a=c}}<big><big>⟩</big></big>}} to Bernoulli numbers:

:<math>\begin{align}
\sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle &= 2^{n+1} (2^{n+1}-1) \frac{B_{n+1}}{n+1}, \\
\sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle \binom{n}{m}^{-1} &= (n+1) B_n.
\end{align}</math>

Both formulae are valid for {{math|''n'' ≥ 0}} if {{math|''B''<sub>1</sub>}} is set to {{sfrac|1|2}}. If {{math|''B''<sub>1</sub>}} is set to −{{sfrac|1|2}} they are valid only for {{math|''n'' ≥ 1}} and {{math|''n'' ≥ 2}} respectively.

==A binary tree representation==
The Stirling polynomials {{math|''σ''<sub>''n''</sub>(''x'')}} are related to the Bernoulli numbers by {{math|''B''<sub>''n''</sub> {{=}} ''n''!''σ''<sub>''n''</sub>(1)}}. S. C. Woon described an algorithm to compute {{math|''σ''<sub>''n''</sub>(1)}} as a binary tree:{{r|Woon1997}}

:]

Woon's recursive algorithm (for {{math|''n'' ≥ 1}}) starts by assigning to the root node {{math|''N'' {{=}} }}. Given a node {{math|''N'' {{=}} }} of the tree, the left child of the node is {{math|''L''(''N'') {{=}} }} and the right child {{math|''R''(''N'') {{=}} }}. A node {{math|''N'' {{=}} }} is written as {{math|±}} in the initial part of the tree represented above with ± denoting the sign of {{math|''a''<sub>1</sub>}}.

Given a node {{mvar|N}} the factorial of {{mvar|N}} is defined as

:<math> N! = a_1 \prod_{k=2}^{\operatorname{length}(N)} a_k!. </math>

Restricted to the nodes {{mvar|N}} of a fixed tree-level {{mvar|n}} the sum of {{math|{{sfrac|1|''N''!}}}} is {{math|''σ''<sub>''n''</sub>(1)}}, thus

:<math> B_n = \sum_\stackrel{N \text{ node of}}{\text{ tree-level } n} \frac{n!}{N!}. </math>

For example:
:{{math|1=''B''<sub>1</sub> = 1!({{sfrac|1|2!}})}}
:{{math|1=''B''<sub>2</sub> = 2!(−{{sfrac|1|3!}} + {{sfrac|1|2!2!}})}}
:{{math|1=''B''<sub>3</sub> = 3!({{sfrac|1|4!}} − {{sfrac|1|2!3!}} − {{sfrac|1|3!2!}} + {{sfrac|1|2!2!2!}})}}


==Integral representation and continuation== ==Integral representation and continuation==


The ] The ]
: <math> b(s) = 2e^{s i \pi/2}\int_0^\infty \frac{st^s}{1-e^{2\pi t}} \frac{dt}{t} = \frac{s!}{2^{s-1}}\frac{\zeta(s)}{{ }\pi^s{ }}(-i)^s= \frac{2s!\zeta(s)}{(2\pi i)^s}</math>
has as special values {{math|''b''(2''n'') {{=}} ''B''<sub>2''n''</sub>}} for {{math|''n'' > 0}}.


For example, {{math|1=''b''(3) = {{sfrac|3|2}}''ζ''(3)''&pi;''<sup>−3</sup>''i''}} and {{math|1=''b''(5) = −{{sfrac|15|2}}''ζ''(5)''&pi;''<sup>−5</sup>''i''}}. Here, {{mvar|ζ}} is the ], and {{mvar|i}} is the ]. Leonhard Euler (''Opera Omnia'', Ser. 1, Vol. 10, p.&nbsp;351) considered these numbers and calculated
: <math> b(s) = 2e^{s i \pi/2}\int_{0}^{\infty} \frac{st^{s}}{1-e^{2\pi t}} \frac{dt}{t} </math>


: <math> \begin{align}
has as special values ''b''(2''n'') = ''B''<sub>2''n''</sub> for ''n''&nbsp;&gt;&nbsp;0. The integral might be considered as a continuation of the Bernoulli numbers to the ] and this was indeed suggested by Peter Luschny in 2004.{{Fact|date=December 2008}}
p &= \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots \right) = 0.0581522\ldots \\
q &= \frac{15}{2\pi^5}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+\cdots \right) = 0.0254132\ldots
\end{align}</math>


Another similar ] representation is
For example ''b''(3) = (3/2)ζ(3)Π<sup>−3</sup>Ι and ''b''(5) = −(15/2)&nbsp;''ζ''(5)&nbsp;Π<sup>&nbsp;−5</sup>Ι. Here ''ζ''(''n'') denotes the ] and Ι the ]. It is remarkable that already Leonhard Euler (''Opera Omnia'', Ser. 1, Vol. 10, p. 351) considered these numbers and calculated
: <math> b(s) = -\frac{e^{s i \pi/2}}{2^{s}-1}\int_0^\infty \frac{st^{s}}{\sinh\pi t} \frac{dt}{t}= \frac{2e^{s i \pi/2}}{2^{s}-1}\int_0^\infty \frac{e^{\pi t}st^s}{1-e^{2\pi t}} \frac{dt}{t}. </math>


==The relation to the Euler numbers and {{pi}}==
: <math> p = \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\text{etc.}\ \right) = 0.0581522\ldots \ , </math>


The ]s are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the
: <math> q = \frac{15}{2\pi^{5}}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+\text{etc.}\ \right) = 0.0254132\ldots \ . </math>
asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers {{math|''E''<sub>2''n''</sub>}} are in magnitude approximately {{math|{{sfrac|2|π}}(4<sup>2''n''</sup> − 2<sup>2''n''</sup>)}} times larger than the Bernoulli numbers {{math|''B''<sub>2''n''</sub>}}. In consequence:


: <math> \pi \sim 2 (2^{2n} - 4^{2n}) \frac{B_{2n}}{E_{2n}}. </math>
Euler's values are unsigned and real, but obviously his aim was to find a meaningful way to define the Bernoulli numbers at the odd integers ''n''&nbsp;&gt;&nbsp;1.


This asymptotic equation reveals that {{pi}} lies in the common root of both the Bernoulli and the Euler numbers. In fact {{pi}} could be computed from these rational approximations.
==The relation to the Euler numbers and ''π'' ==


Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd {{mvar|n}}, {{math|''B''<sub>''n''</sub> {{=}} ''E''<sub>''n''</sub> {{=}} 0}} (with the exception {{math|''B''<sub>1</sub>}}), it suffices to consider the case when {{mvar|n}} is even.
The ] are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the
asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers ''E''<sub>2''n''</sub> are in magnitude approximately (2/π)(4<sup>2''n''</sup>&nbsp;−&nbsp;2<sup>2''n''</sup>) times larger than the Bernoulli numbers ''B''<sub>2''n''</sub>. In consequence:


:<math>\begin{align}
: <math> \pi \ \sim \ 2 \left(2^{2n} - 4^{2n} \right) \frac{B_{2n}}{E_{2n}} \ . </math>
B_n &= \sum_{k=0}^{n-1}\binom{n-1}{k} \frac{n}{4^n-2^n}E_k & n&=2, 4, 6, \ldots \\
E_n &= \sum_{k=1}^n \binom{n}{k-1} \frac{2^k-4^k}{k} B_k & n&=2,4,6,\ldots
\end{align}</math>


These conversion formulas express a connection between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to {{pi}}. These numbers are defined for {{math|''n'' ≥ 1}} as<ref>{{citation
This asymptotic equation reveals that ''π'' lies in the common root of both the Bernoulli and the Euler numbers. In fact ''π'' could be computed from these rational approximations.
| last = Stanley | first = Richard P. | author-link = Richard P. Stanley
| arxiv = 0912.4240
| contribution = A survey of alternating permutations
| doi = 10.1090/conm/531/10466
| mr = 2757798
| pages = 165–196
| publisher = American Mathematical Society | location = Providence, RI
| series = Contemporary Mathematics
| title = Combinatorics and graphs
| volume = 531
| year = 2010| isbn = 978-0-8218-4865-4 | s2cid = 14619581 }}</ref>{{r|Elkies2003}}


:<math> S_n = 2 \left(\frac{2}{\pi}\right)^n \sum_{k = 0}^\infty \frac{ (-1)^{kn} }{(2k+1)^n} = 2 \left(\frac{2}{\pi}\right)^n \lim_{K\to \infty} \sum_{k = -K}^K (4k+1)^{-n}. </math>
Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since for ''n'' odd ''B''<sub>''n''</sub> =&nbsp;''E''<sub>''n''</sub> = 0 (with the exception ''B''<sub>1</sub>), it suffices to regard the case when ''n'' is even.


The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by ] in a landmark paper ''De summis serierum reciprocarum'' (On the sums of series of reciprocals) and has fascinated mathematicians ever since.{{r|Euler1735}} The first few of these numbers are
: <math> B_{n}=\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{n}{4^n-2^n}E_{k} \quad (n=2,4,6,\ldots) </math>


: <math> E_{n}=\sum_{k=1}^{n}\binom{n}{k-1}\frac{2^k-4^k}{k} B_{k} \quad (n=2,4,6,\ldots). </math> : <math> S_n = 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots </math> ({{OEIS2C|id=A099612}} / {{OEIS2C|id=A099617}})


These are the coefficients in the expansion of {{math|sec ''x'' + tan ''x''}}.
These conversion formulas express an ] between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to ''π''. These numbers are defined for ''n'' > 1 as


The Bernoulli numbers and Euler numbers can be understood as ''special views'' of these numbers, selected from the sequence {{math|''S''<sub>''n''</sub>}} and scaled for use in special applications.
: <math> S_n = 2 \left(\frac{2}{\pi}\right)^{n}\sum_{k=-\infty}^{\infty}\left(4k+1\right)^{-n} \quad (k=0,-1,1,-2,2,\ldots) </math>


: <math>\begin{align}
and ''S''<sub>1</sub> = 1 by convention. The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by ] 1734 in a landmark paper `De summis serierum reciprocarum' (On the sums of series of reciprocals) and fascinated mathematicians ever since. The first few of these numbers are
B_{n} &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} \frac{n! }{2^n - 4^n}\, S_{n}\ , & n&= 2, 3, \ldots \\
E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} n! \, S_{n+1} & n &= 0, 1, \ldots
\end{align}</math>


The expression has the value 1 if {{math|''n''}} is even and 0 otherwise (]).
: <math> 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots </math>


These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of {{math|''R''<sub>''n''</sub> {{=}} {{sfrac|2''S''<sub>''n''</sub>|''S''<sub>''n'' + 1</sub>}}}} when {{mvar|n}} is even. The {{math|''R''<sub>''n''</sub>}} are rational approximations to {{pi}} and two successive terms always enclose the true value of {{pi}}. Beginning with {{math|''n'' {{=}} 1}} the sequence starts ({{OEIS2C|id=A132049}} / {{OEIS2C|id=A132050}}):
The Bernoulli numbers and Euler numbers are best understood as ''special views'' of these numbers, selected from the sequence ''S''<sub>''n''</sub> and scaled for use in special applications.


: <math> B_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left \frac{n! }{2^n-4^n}\, S_{n}\ , \quad (n=2,3,\ldots) \ , </math> : <math> 2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385}, \frac{12465}{3968}, \frac{158720}{50521},\ldots \quad \longrightarrow \pi. </math>


These rational numbers also appear in the last paragraph of Euler's paper cited above.
: <math> E_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left n! \, S_{n+1} \quad\qquad (n=0,1,\ldots) \ . </math>


Consider the Akiyama–Tanigawa transform for the sequence {{OEIS2C|id=A046978}} ({{math|''n'' + 2}}) / {{OEIS2C|id=A016116}} ({{math|''n'' + 1}}):
The expression has the value 1 if ''n'' is even and 0 otherwise (]).


:{| class="wikitable" style="text-align:right;"
These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of ''R''<sub>n</sub> = 2 ''S''<sub>''n''</sub> / ''S''<sub>''n''+1</sub> when ''n'' is even. The ''R''<sub>''n''</sub> are rational approximations to ''π'' and two successive terms always enclose the true value of ''π''. Beginning with ''n'' = 1 the sequence starts
! 0

|1||{{sfrac|1|2}}||0||−{{sfrac|1|4}}||−{{sfrac|1|4}}||−{{sfrac|1|8}}||0
: <math> 2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385},\ldots \quad \longrightarrow \pi \ . </math>
|-
! 1
| {{sfrac|1|2}}|| 1|| {{sfrac|3|4}}|| 0|| −{{sfrac|5|8}}|| −{{sfrac|3|4}}||
|-
! 2
| −{{sfrac|1|2}}|| {{sfrac|1|2}}|| {{sfrac|9|4}}|| {{sfrac|5|2}}|| {{sfrac|5|8}}|| ||
|-
! 3
| −1|| −{{sfrac|7|2}}|| −{{sfrac|3|4}}|| {{sfrac|15|2}}|| || ||
|-
! 4
| {{sfrac|5|2}}|| −{{sfrac|11|2}}|| −{{sfrac|99|4}}|| || || ||
|-
! 5
| 8|| {{sfrac|77|2}}|| || || || ||
|-
! 6
| −{{sfrac|61|2}}|| || || || || ||
|}


From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −{{sfrac|1|2}} × {{OEIS2C|id=A163982}}.
These rational numbers also appear in the last paragraph of Euler's paper cited above. But it was only in September 2007 that this classical sequence found its way into the Encyclopedia of Integer Sequences (A132049).


==An algorithmic view: the Seidel triangle== ==An algorithmic view: the Seidel triangle==


The sequence ''S''<sub>''n''</sub> has another unexpected yet important property: The denominators of ''S''<sub>''n''</sub> divide the factorial (''n''&nbsp;−&nbsp;1)<nowiki>!</nowiki>. In other words: the numbers ''T''<sub>''n''</sub>&nbsp;=&nbsp;''S''<sub>''n''</sub>(''n''&nbsp;−&nbsp;1)! are integers. The sequence ''S''<sub>''n''</sub> has another unexpected yet important property: The denominators of ''S''<sub>''n''+1</sub> divide the factorial {{math|''n''!}}. In other words: the numbers {{math|1=''T''<sub>''n''</sub>&nbsp;=&nbsp;''S''<sub>''n'' + 1</sub> ''n''!}}, sometimes called ], are integers.


: <math> T_{n} = 1,1,1,2,5,16,61,272,1385,7936,50521,353792,\ldots \quad (n=1,2,\ldots) </math> : <math> T_n = 1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0, 1, 2, 3, \ldots </math> ({{OEIS2C|id=A000111}}). See ({{OEIS2C|id=A253671}}).

Their ] is the sum of the ] and ] functions.

: <math> \sum_{n=0}^\infty T_n \frac{x^n}{n!} = \tan \left(\frac\pi4 + \frac x2\right) = \sec x + \tan x</math>.


Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as


:<math>\begin{align}
: <math> B_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left \frac{n }{2^n-4^n}\, T_{n}\ , \quad (n=2,3,\ldots) \ , </math>
B_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} \frac{n }{2^n-4^n}\, T_{n-1}\ & n &\geq 2 \\
E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} T_{n} & n &\geq 0
\end{align}</math>


These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers {{math|''E''<sub>2''n''</sub>}} are given immediately by {{math|''T''<sub>2''n''</sub>}} and the Bernoulli numbers {{math|''B''<sub>2''n''</sub>}} are fractions obtained from {{math|''T''<sub>2''n'' - 1</sub>}} by some easy shifting, avoiding rational arithmetic.
: <math> E_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left T_{n+1} \quad\quad\qquad(n=0,1,\ldots) \ . </math>


What remains is to find a convenient way to compute the numbers {{math|''T''<sub>''n''</sub>}}. However, already in 1877 ] published an ingenious algorithm, which makes it simple to calculate {{math|''T''<sub>''n''</sub>}}.{{r|Seidel1877}}
These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers ''E''<sub>''n''</sub> are given immediately by ''T''<sub>2''n''&nbsp;+&nbsp;1</sub> and the Bernoulli numbers ''B''<sub>2''n''</sub> are obtained from ''T''<sub>2''n''</sub> by some easy shifting, avoiding rational arithmetic.


{{image frame|align=none|caption=Seidel's algorithm for {{math|''T''<sub>''n''</sub>}}
What remains is to find a convenient way to compute the numbers ''T''<sub>''n''</sub>. However, already in 1877 ] published an ingenious algorithm which makes it extremely simple to calculate ''T''<sub>''n''</sub>.
|content=<math>
\begin{array}{crrrcc}
{ } & { } & {\color{red}1} & { } & { } & { } \\
{ } & {\rightarrow} & {\color{blue}1} & {\color{red}1} & { } \\
{ } & {\color{red}2} & {\color{blue}2} & {\color{blue}1} & {\leftarrow} \\
{\rightarrow} & {\color{blue}2} & {\color{blue}4} & {\color{blue}5} & {\color{red}5} \\
{\color{red}16} & {\color{blue}16} & {\color{blue}14} & {\color{blue}10} & {\color{blue}5} & {\leftarrow}
\end{array}
</math>}}
#Start by putting 1 in row 0 and let {{math|''k''}} denote the number of the row currently being filled
#If {{math|''k''}} is odd, then put the number on the left end of the row {{math|''k'' − 1}} in the first position of the row {{math|''k''}}, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper
#At the end of the row duplicate the last number.
#If {{math|''k''}} is even, proceed similar in the other direction.


Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont {{r|Dumont1981}}) and was rediscovered several times thereafter.
<center><table >
<tr style="padding: 1.5em; line-height: 1.5em; background:#D9DECD; color:red; text-align:center">
<td>Seidel's algorithm for T<sub>''n''</sub></td></tr><tr><td>
]
</td></tr></table></center>


Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers {{math|''T''<sub>2''n''</sub>}} and recommended this method for computing {{math|''B''<sub>2''n''</sub>}} and {{math|''E''<sub>2''n''</sub>}} 'on electronic computers using only simple operations on integers'.{{r|KnuthBuckholtz1967}}
Start by putting 1 in row 0 and let ''k'' denote the number of the row currently being filled. If ''k'' is odd, then put the number on the left end of the row ''k''&nbsp;−&nbsp;1 in the first position of the row ''k'', and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper. At the end of the row duplicate the last number. If ''k'' is even, proceed similar in the other direction.


V. I. Arnold{{r|Arnold1991}} rediscovered Seidel's algorithm and later Millar, Sloane and Young popularized Seidel's algorithm under the name ].
Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont (1981)) and was rediscovered several times thereafter.


Triangular form:
Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz (1967) gave a recurrence equation for the numbers ''T''<sub>2''n''</sub> and recommended this method for computing ''B''<sub>2''n''</sub> and ''E''<sub>2''n''</sub> ‘on electronic computers using only simple operations on integers’.


:{| style="text-align:right"
V. I. Arnold rediscovered Seidel's algorithm in 1991 and later Millar, Sloane and Young popularized Seidel's algorithm under the name ].
| || || || || || || 1|| || || || || ||
|-
| || || || || || 1|| || 1|| || || || ||
|-
| || || || || 2|| || 2|| || 1|| || || ||
|-
| || || || 2|| || 4|| || 5|| || 5|| || ||
|-
| || || 16|| || 16|| || 14|| || 10|| || 5|| ||
|-
| || 16|| || 32|| || 46|| || 56|| || 61|| || 61||
|-
|272|| ||272|| ||256|| ||224|| ||178|| ||122|| || 61
|}

Only {{OEIS2C|id=A000657}}, with one 1, and {{OEIS2C|id=A214267}}, with two 1s, are in the OEIS.

Distribution with a supplementary 1 and one 0 in the following rows:

:{| style="text-align:right"
| || || || || || || 1|| || || || || ||
|-
| || || || || || 0|| || 1|| || || || ||
|-
| || || || || −1|| || −1|| || 0|| || || ||
|-
| || || || 0|| || −1|| || −2|| || −2|| || ||
|-
| || || 5|| || 5|| || 4|| || 2|| || 0|| ||
|-
| || 0|| || 5|| || 10|| || 14|| || 16|| || 16||
|-
|−61|| ||−61|| ||−56|| ||−46|| ||−32|| ||−16|| || 0
|}

This is {{OEIS2C|id=A239005}}, a signed version of {{OEIS2C|id=A008280}}. The main andiagonal is {{OEIS2C|id=A122045}}. The main diagonal is {{OEIS2C|id=A155585}}. The central column is {{OEIS2C|id=A099023}}. Row sums: 1, 1, −2, −5, 16, 61.... See {{OEIS2C|id=A163747}}. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below.

The Akiyama–Tanigawa algorithm applied to {{OEIS2C|id=A046978}} ({{math|''n'' + 1}}) / {{OEIS2C|id=A016116}}({{math|''n''}}) yields:

:{| style="text-align:right"
| 1|| 1|| {{sfrac|1|2}}|| 0|| −{{sfrac|1|4}}|| −{{sfrac|1|4}}|| −{{sfrac|1|8}}
|-
| 0|| 1|| {{sfrac|3|2}}|| 1|| 0|| −{{sfrac|3|4}}
|-
| −1|| −1|| {{sfrac|3|2}}|| 4|| {{sfrac|15|4}}
|-
| 0|| −5|| −{{sfrac|15|2}}|| 1
|-
| 5|| 5|| −{{sfrac|51|2}}
|-
| 0|| 61
|-
| −61
|}

'''1.''' The first column is {{OEIS2C|id=A122045}}. Its binomial transform leads to:

:{| style="text-align:right"
|-
| 1|| 1|| 0|| −2|| 0|| 16|| 0
|-
|0||−1||−2||2||16||−16
|-
|−1||−1||4||14||−32
|-
|0||5||10||−46
|-
|5||5||−56
|-
|0||−61
|-
|−61
|}

The first row of this array is {{OEIS2C|id=A155585}}. The absolute values of the increasing antidiagonals are {{OEIS2C|id=A008280}}. The sum of the antidiagonals is {{nowrap|−{{OEIS2C|id=A163747}} ({{math|''n'' + 1}}).}}

'''2.''' The second column is {{nowrap|1 1 −1 −5 5 61 −61 −1385 1385...}}. Its binomial transform yields:

:{| style="text-align:right"
|-
| 1|| 2|| 2|| −4|| −16|| 32|| 272
|-
|1||0||−6||−12||48||240
|-
|−1||−6||−6||60||192
|-
|−5||0||66||32
|-
|5||66||66
|-
|61||0
|-
|−61
|}

The first row of this array is {{nowrap|1 2 2 −4 −16 32 272 544 −7936 15872 353792 −707584...}}. The absolute values of the second bisection are the double of the absolute values of the first bisection.

Consider the Akiyama-Tanigawa algorithm applied to {{OEIS2C|id=A046978}} ({{math|''n''}}) / ({{OEIS2C|id=A158780}} ({{math|''n'' + 1}}) =&nbsp;abs({{OEIS2C|id=A117575}} ({{mvar|n}})) + 1 = {{nowrap|1, 2, 2, {{sfrac|3|2}}, 1, {{sfrac|3|4}}, {{sfrac|3|4}}, {{sfrac|7|8}}, 1, {{sfrac|17|16}}, {{sfrac|17|16}}, {{sfrac|33|32}}...}}.

:{| style="text-align:right"
|1||2||2||{{sfrac|3|2}}||1||{{sfrac|3|4}}||{{sfrac|3|4}}
|-
|−1||0||{{sfrac|3|2}}||2||{{sfrac|5|4}}||0
|-
|−1||−3||−{{sfrac|3|2}}||3||{{sfrac|25|4}}
|-
|2||−3||−{{sfrac|27|2}}||−13
|-
|5||21||−{{sfrac|3|2}}
|-
|−16||45
|-
|−61
|}

The first column whose the absolute values are {{OEIS2C|id=A000111}} could be the numerator of a trigonometric function.

{{OEIS2C|id=A163747}} is an autosequence of the first kind (the main diagonal is {{OEIS2C|id=A000004}}). The corresponding array is:

:{| style="text-align:right"
|0||−1||−1||2||5||−16||−61
|-
|−1||0||3||3||−21||−45
|-
|1||3||0||−24||−24
|-
|2||−3||−24||0
|-
|−5||−21||24
|-
|−16||45
|-
|−61
|}

The first two upper diagonals are {{nowrap|−1 3 −24 402...}} = {{math|(−1)<sup>''n'' + 1</sup>}}&nbsp;×&nbsp;{{OEIS2C|id=A002832}}. The sum of the antidiagonals is {{nowrap|0 −2 0 10...}} = 2&nbsp;×&nbsp;{{OEIS2C|id=A122045}}(''n''&nbsp;+&nbsp;1).

−{{OEIS2C|id=A163982}} is an autosequence of the second kind, like for instance {{OEIS2C|id=A164555}} / {{OEIS2C|id=A027642}}. Hence the array:

:{| style="text-align:right"
|-
|2||1||−1||−2||5||16||−61
|-
|−1||−2||−1||7||11||−77
|-
|−1||1||8||4||−88
|-
|2||7||−4||−92
|-
|5||−11||−88
|-
|−16||−77
|-
|−61
|}

The main diagonal, here {{nowrap|2 −2 8 −92...}}, is the double of the first upper one, here {{OEIS2C|id=A099023}}. The sum of the antidiagonals is {{nowrap|2 0 −4 0...}} = 2&nbsp;×&nbsp;{{OEIS2C|id=A155585}}({{math|''n'' + }}1). {{OEIS2C|id=A163747}}&nbsp;−&nbsp;{{OEIS2C|id=A163982}} =&nbsp;2&nbsp;×&nbsp;{{OEIS2C|id=A122045}}.


==A combinatorial view: alternating permutations== ==A combinatorial view: alternating permutations==
{{main|Alternating permutations}} {{main|Alternating permutations}}


Around 1880, three years after the publication of Seidel's algorithm, ] proved a now classic result of combinatorial analysis. Looking at the first terms of the Taylor expansion of the ] Around 1880, three years after the publication of Seidel's algorithm, ] proved a now classic result of combinatorial analysis.{{r|André1879|André1881}} Looking at the first terms of the Taylor expansion of the ]
tan ''x'' and sec ''x'' André made a startling discovery. {{math|tan ''x''}} and {{math|sec ''x''}} André made a startling discovery.


:<math>\begin{align}
: <math> \tan x = 1\frac{x}{1!} + 2\frac{x^3}{3!} + 16\frac{x^5}{5!} + 272\frac{x^7}{7!} + 7936\frac{x^9}{9!} + \cdots </math>
\tan x &= x + \frac{2x^3}{3!} + \frac{16x^5}{5!} + \frac{272x^7}{7!} + \frac{7936x^9}{9!} + \cdots\\
\sec x &= 1 + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} + \frac{50521x^{10}}{10!} + \cdots
\end{align}</math>


The coefficients are the ]s of odd and even index, respectively. In consequence the ordinary expansion of {{math|tan ''x'' + sec ''x''}} has as coefficients the rational numbers {{math|''S''<sub>''n''</sub>}}.
: <math> \sec x = 1 + 1\frac{x^2}{2!} + 5\frac{x^4}{4!} + 61\frac{x^6}{6!} + 1385\frac{x^8}{8!} + 50521\frac{x^{10}}{10!} + \cdots </math>


: <math> \tan x + \sec x = 1 + x + \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 + \tfrac{5}{24}x^4 + \tfrac{2}{15}x^5 + \tfrac{61}{720}x^6 + \cdots </math>
The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of tan&nbsp;''x''&nbsp;+&nbsp;sec&nbsp;''x'' has as coefficients the rational numbers ''S''<sub>''n''</sub>.


André then succeeded by means of a recurrence argument to show that the ]s of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).
: <math> \tan x + \sec x = 1 + 1x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \frac{5}{24}x^4 + \frac{2}{15}x^5 + \frac{61}{720}x^6 + \cdots </math>


==Related sequences==
André then succeeded by means of a recurrence argument to show that the ] of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).
The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers:
{{math|1=''B''<sub>0</sub> = 1}}, {{math|1=''B''<sub>1</sub> = 0}}, {{math|1=''B''<sub>2</sub> = {{sfrac|1|6}}}}, {{math|1=''B''<sub>3</sub> = 0}}, {{math|1=''B''<sub>4</sub> = −{{sfrac|1|30}}}}, {{OEIS2C|id=A176327}} / {{OEIS2C|id=A027642}}. Via the second row of its inverse Akiyama–Tanigawa transform {{OEIS2C|id=A177427}}, they lead to Balmer series {{OEIS2C|id=A061037}} / {{OEIS2C|id=A061038}}.


The Akiyama–Tanigawa algorithm applied to {{OEIS2C|id=A060819}} ({{math|''n'' + 4}}) / {{OEIS2C|id=A145979}} ({{mvar|n}}) leads to the Bernoulli numbers {{OEIS2C|id=A027641}} / {{OEIS2C|id=A027642}}, {{OEIS2C|id=A164555}} / {{OEIS2C|id=A027642}}, or {{OEIS2C|id=A176327}} {{OEIS2C|id=A176289}} without {{math|''B''<sub>1</sub>}}, named intrinsic Bernoulli numbers {{math|''B''<sub>''i''</sub>(''n'')}}.
==Generalizations by polynomials==


:{| style="text-align:center; padding-left; padding-right: 2em;"
The ] can be regarded as generalizations of the Bernoulli numbers the same as the Euler polynomials are generalizations of the Euler numbers.
|-
|1||{{sfrac|5|6}}||{{sfrac|3|4}}||{{sfrac|7|10}}||{{sfrac|2|3}}
|-
|{{sfrac|1|6}}||{{sfrac|1|6}}||{{sfrac|3|20}}||{{sfrac|2|15}}||{{sfrac|5|42}}
|-
|0||{{sfrac|1|30}}||{{sfrac|1|20}}||{{sfrac|2|35}}||{{sfrac|5|84}}
|-
|−{{sfrac|1|30}}||−{{sfrac|1|30}}||−{{sfrac|3|140}}||−{{sfrac|1|105}}||0
|-
|0||−{{sfrac|1|42}}||−{{sfrac|1|28}}||−{{sfrac|4|105}}||−{{sfrac|1|28}}
|}

Hence another link between the intrinsic Bernoulli numbers and the Balmer series via {{OEIS2C|id=A145979}} ({{math|''n''}}).

{{OEIS2C|id=A145979}} ({{math|''n'' − 2}}) = 0, 2, 1, 6,... is a permutation of the non-negative numbers.

The terms of the first row are f(n) = {{math|{{sfrac|1|2}} + {{sfrac|1|''n'' + 2}}}}. 2, f(n) is an autosequence of the second kind. 3/2, f(n) leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5 ... = 1/2 + log 2.

Consider g(n) = 1/2 – 1 / (n+2) = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives:

:{| style="text-align:center; padding-left; padding-right:2em;"
|-
|0||{{sfrac|1|6}}||{{sfrac|1|4}}||{{sfrac|3|10}}||{{sfrac|1|3}}||{{sfrac|5|14}}||...
|-
|−{{sfrac|1|6}}||−{{sfrac|1|6}}||−{{sfrac|3|20}}||−{{sfrac|2|15}}||−{{sfrac|5|42}}||−{{sfrac|3|28}}||...
|-
|0||−{{sfrac|1|30}}||−{{sfrac|1|20}}||−{{sfrac|2|35}}||−{{sfrac|5|84}}||−{{sfrac|5|84}}||...
|-
|{{sfrac|1|30}}||{{sfrac|1|30}}||{{sfrac|3|140}}||{{sfrac|1|105}}||0||−{{sfrac|1|140}}||...
|}

0, g(n), is an autosequence of the second kind.

Euler {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}) without the second term ({{sfrac|1|2}}) are the fractional intrinsic Euler numbers {{math|''E''<sub>''i''</sub>(''n'') {{=}} 1, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, 0, −{{sfrac|17|8}}, 0, ...}} The corresponding Akiyama transform is:

:{| style="text-align:center; padding-left; padding-right: 2em;"
|-
|1||1||{{sfrac|7|8}}||{{sfrac|3|4}}||{{sfrac|21|32}}
|-
|0||{{sfrac|1|4}}||{{sfrac|3|8}}||{{sfrac|3|8}}||{{sfrac|5|16}}
|-
|−{{sfrac|1|4}}||−{{sfrac|1|4}}||0||{{sfrac|1|4}}||{{sfrac|25|64}}
|-
|0||−{{sfrac|1|2}}||−{{sfrac|3|4}}||−{{sfrac|9|16}}||−{{sfrac|5|32}}
|-
|{{sfrac|1|2}}||{{sfrac|1|2}}||−{{sfrac|9|16}}||−{{sfrac|13|8}}||−{{sfrac|125|64}}
|}

The first line is {{math|''Eu''(''n'')}}. {{math|''Eu''(''n'')}} preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are {{OEIS2C|id=A069834}} preceded by 0. The difference table is:

:{| style="text-align:center; padding-left; padding-right: 2em;"
|-
|0||1||1||{{sfrac|7|8}}||{{sfrac|3|4}}||{{sfrac|21|32}}||{{sfrac|19|32}}
|-
|1||0||−{{sfrac|1|8}}||−{{sfrac|1|8}}||−{{sfrac|3|32}}||−{{sfrac|1|16}}||−{{sfrac|5|128}}
|-
|−1||−{{sfrac|1|8}}||0||{{sfrac|1|32}}||{{sfrac|1|32}}||{{sfrac|3|128}}||{{sfrac|1|64}}
|}


==Arithmetical properties of the Bernoulli numbers== ==Arithmetical properties of the Bernoulli numbers==


The Bernoulli numbers can be expressed in terms of the Riemann zeta function as ''B''<sub>''n''</sub> = − ''n''ζ(1 − ''n'') for integers ''n'' ≥ 0 provided for ''n'' = 0 and ''n'' = 1 the expression − ''n''ζ(1 − ''n'') is understood as the limiting value and the convention B<sub>1</sub> = 1/2 is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties, a fact discovered by ] in his work on ]. The Bernoulli numbers can be expressed in terms of the Riemann zeta function as {{math|''B''<sub>''n''</sub> {{=}} −''nζ''(1 − ''n'')}} for integers {{math|''n'' ≥ 0}} provided for {{math|''n'' {{=}} 0}} the expression {{math|−''nζ''(1 − ''n'')}} is understood as the limiting value and the convention {{math|''B''<sub>1</sub> {{=}} {{sfrac|1|2}}}} is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the ] postulates that {{mvar|p}} is a prime number if and only if {{math|''pB''<sub>''p'' − 1</sub>}} is congruent to −1 modulo {{mvar|p}}. Divisibility properties of the Bernoulli numbers are related to the ]s of ]s by a theorem of Kummer and its strengthening in the ], and to class numbers of real quadratic fields by ].


=== The Kummer theorems ===
Divisibility properties of the Bernoulli numbers are related to the ]s of ]s by a theorem of Kummer and its strengthening in the ], and to class numbers of real quadratic fields by ]. We also have a relationship to ]; if ''c''<sub>''n''</sub> is the numerator of ''B''<sub>''n''</sub>/2''n'', then the order of <math>K_{4n-2}(\Bbb{Z})</math> is −''c''<sub>2''n''</sub> if ''n'' is even, and 2''c''<sub>2''n''</sub> if ''n'' is odd.
The Bernoulli numbers are related to ] (FLT) by ]'s theorem,{{r|Kummer1850}} which says:


:If the odd prime {{mvar|p}} does not divide any of the numerators of the Bernoulli numbers {{math|''B''<sub>2</sub>, ''B''<sub>4</sub>, ..., ''B''<sub>''p'' − 3</sub>}} then {{math|''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> + ''z''<sup>''p''</sup> {{=}} 0}} has no solutions in nonzero integers.
The ] postulates that ''p'' is a prime number if and only if ''pB''<sub>''p''−1</sub> is congruent to −1 mod ''p''.


Prime numbers with this property are called ]s. Another classical result of Kummer are the following ].{{r|Kummer1851}}
=== Von Staudt–Clausen theorem ===


{{main|Kummer's congruence}}
The von Staudt–Clausen theorem was given by ] and ] independently in 1840. It describes the arithmetical structure of the Bernoulli numbers.


:Let {{mvar|p}} be an odd prime and {{mvar|b}} an even number such that {{math|''p''&nbsp;−&nbsp;1}} does not divide {{mvar|b}}. Then for any non-negative integer {{mvar|k}}
The von Staudt–Clausen theorem has two parts. The first one describes how the denominators of the Bernoulli numbers can be computed. Paraphrasing the words of Clausen it can be stated as:
:: <math> \frac{B_{k(p-1)+b}}{k(p-1)+b} \equiv \frac{B_{b}}{b} \pmod{p}. </math>


A generalization of these congruences goes by the name of {{math|''p''}}-adic continuity.
<p style="margin-left:40px">“The denominator of the 2''n''th Bernoulli number can be found as follows: Add to all divisors of 2''n'', 1,&nbsp;2,&nbsp;''a'',&nbsp;''a''<nowiki>'</nowiki>,&nbsp;...,&nbsp;2''n'' the unity, which gives the sequence 2,&nbsp;3,&nbsp;''a''&nbsp;+&nbsp;1,&nbsp;''a''<nowiki>'</nowiki>&nbsp;+&nbsp;1,&nbsp;...,&nbsp;2''n''&nbsp;+&nbsp;1. Select from this sequence only the prime numbers 2,&nbsp;3,&nbsp;''p'',&nbsp;''p''<nowiki>'</nowiki>, etc. and build their product.”</p>


==={{math|''p''}}-adic continuity===
Clausen's algorithm translates almost verbatim to a modern computer algebra program, which looks similar to the pseudocode on the left hand site of the following table. On the right hand side the computation is traced for the input ''n''&nbsp;=&nbsp;88. It shows that the denominator of ''B''<sub>88</sub> is 61410.&nbsp;


If {{mvar|b}}, {{mvar|m}} and {{mvar|n}} are positive integers such that {{mvar|m}} and {{mvar|n}} are not divisible by {{math|''p'' − 1}} and {{math|''m'' ≡ ''n'' (mod ''p''<sup>''b'' − 1</sup> (''p'' − 1))}}, then
<center><table style="border: 3px solid #D9DECD;" >
<tr style="padding: 1.5em; line-height: 1.5em; background:#D9DECD; color:red; text-align:center">
<td colspan="3" >Clausen's algorithm for the denominator of B<sub>''n''</sub></td></tr>
<tr>
<td>Clausen: function(integer ''n'')</td>
<td>&nbsp;&nbsp;|&nbsp;&nbsp;</td>
<td>''n'' = 88</td>
</tr> <tr>
<td>''S'' = divisors(''n'');</td>
<td>&nbsp;&nbsp;|&nbsp;&nbsp;</td>
<td>{1, 2, 4, 8, 11, 22, 44, 88}</td>
</tr> <tr>
<td>S = map(''k'' → ''k''&nbsp;+&nbsp;1, ''S'');</td>
<td>&nbsp;&nbsp;|&nbsp;&nbsp;</td>
<td>{2, 3, 5, 9, 12, 23, 45, 89}</td>
</tr> <tr>
<td>''S'' = select(isprime, ''S'');</td>
<td>&nbsp;&nbsp;|&nbsp;&nbsp;</td>
<td>{2, 3, 5, 23, 89}</td>
</tr> <tr>
<td>return product(''S'');</td>
<td>&nbsp;&nbsp;|&nbsp;&nbsp;</td>
<td>61410</td>
</tr>
</table></center>


:<math>(1-p^{m-1})\frac{B_m}{m} \equiv (1-p^{n-1})\frac{B_n} n \pmod{p^b}.</math>
The second part of the von Staudt–Clausen theorem is a very remarkable representation of the Bernoulli numbers. This representation is given for the first few nonzero Bernoulli numbers in the next table.


Since {{math|''B''<sub>''n''</sub> {{=}} −''nζ''(1 − ''n'')}}, this can also be written
<center>
<table style="border: 3px solid #D9DECD;" >
<tr style="padding: 1.5em; line-height: 1.5em; background:#D9DECD; color:red; text-align:center">
<td colspan="3" >Von Staudt–Clausen representation of B<sub>''n''</sub></td></tr>
<tr>
<td>''B''<sub>0</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1</td>
</tr> <tr>
<td>''B''<sub>1</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1&nbsp;−&nbsp;1/2</td>
</tr> <tr>
<td>''B''<sub>2</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1&nbsp;−&nbsp;1/2&nbsp;−&nbsp;1/3</td>
</tr> <tr>
<td>''B''<sub>4</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1&nbsp;−&nbsp;1/2&nbsp;−&nbsp;1/3&nbsp;−&nbsp;1/5</td>
</tr> <tr>
<td>''B''<sub>6</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1&nbsp;−&nbsp;1/2&nbsp;−&nbsp;1/3&nbsp;−&nbsp;1/7</td>
</tr> <tr>
<td>''B''<sub>8</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1&nbsp;−&nbsp;1/2&nbsp;−&nbsp;1/3&nbsp;−&nbsp;1/5</td>
</tr> <tr>
<td>''B''<sub>10</sub></td>
<td>&nbsp;=&nbsp;</td>
<td>1&nbsp;−&nbsp;1/2&nbsp;−&nbsp;1/3&nbsp;−&nbsp;1/11</td>
</tr>
</table></center>


:<math>\left(1-p^{-u}\right)\zeta(u) \equiv \left(1-p^{-v}\right)\zeta(v) \pmod{p^b},</math>
The theorem affirms the existence of an integer ''I''<sub>n</sub> such that


where {{math|''u'' {{=}} 1 − ''m''}} and {{math|''v'' {{=}} 1 − ''n''}}, so that {{mvar|u}} and {{mvar|v}} are nonpositive and not congruent to 1 modulo {{math|''p'' − 1}}. This tells us that the Riemann zeta function, with {{math|1 − ''p''<sup>−''s''</sup>}} taken out of the Euler product formula, is continuous in the ]s on odd negative integers congruent modulo {{math|''p'' − 1}} to a particular {{math|''a'' ≢ 1 mod (''p'' − 1)}}, and so can be extended to a continuous function {{math|''ζ''<sub>''p''</sub>(''s'')}} for all {{mvar|p}}-adic integers <math>\mathbb{Z}_p,</math> the ].
: <math> B_{n}=\frac{I_n}{2} - \sum_{(p-1)|n} \frac1p \quad (n\geq0)\ . </math>


=== Ramanujan's congruences ===
The sum is over the primes ''p'' for which ''p''&nbsp;−&nbsp;1 divides ''n''. These are the same primes which are employed in the Clausen algorithm. The proposition holds true for ''all'' ''n''&nbsp;≥&nbsp;0, not only for even ''n''. ''I''<sub>1</sub>&nbsp;=&nbsp;2 and for odd ''n''&nbsp;&gt;&nbsp;1, ''I''<sub>''n''</sub>&nbsp;=&nbsp;1.

The following relations, due to ], provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition:

:<math>\binom{m+3}{m} B_m=\begin{cases}
\frac{m+3}{3}-\sum\limits_{j=1}^\frac{m}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 0\pmod 6;\\
\frac{m+3}{3}-\sum\limits_{j=1}^\frac{m-2}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 2\pmod 6;\\
-\frac{m+3}{6}-\sum\limits_{j=1}^\frac{m-4}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 4\pmod 6.\end{cases}</math>

=== Von Staudt–Clausen theorem ===
{{main|Von Staudt–Clausen theorem}}
The von Staudt–Clausen theorem was given by ]{{r|vonStaudt1840}} and ]{{r|Clausen1840}} independently in 1840. The theorem states that for every {{math|''n'' > 0}},
: <math> B_{2n} + \sum_{(p-1)\,\mid\,2n} \frac1p</math>
is an integer. The sum extends over all ] {{math|''p''}} for which {{math|''p'' − 1}} divides {{math|2''n''}}.


A consequence of this is that the denominator of {{math|''B''<sub>2''n''</sub>}} is given by the product of all primes {{math|''p''}} for which {{math|''p'' − 1}} divides {{math|2''n''}}. In particular, these denominators are ] and divisible by&nbsp;6.
Consequences of the von Staudt–Clausen theorem are: the denominators of the Bernoulli numbers are square-free and for ''n'' ≥ 2 divisible by 6.


=== Why do the odd Bernoulli numbers vanish? === === Why do the odd Bernoulli numbers vanish? ===
The sum The sum


:<math>\varphi_k(n) = \sum_{i=0}^n i^k - \frac{n^k}{2}</math> :<math>\varphi_k(n) = \sum_{i=0}^n i^k - \frac{n^k} 2</math>


can be evaluated for negative values of the index ''n''. Doing so will show that it is an ] for even values of ''k'', which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that ''B''<sub>2''k''+1−''m''</sub> is 0 for ''m'' odd and greater than 1; and that the term for ''B''<sub>1</sub> is cancelled by the subtraction. The von Staudt-Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for ''n''&nbsp;&gt;&nbsp;1). can be evaluated for negative values of the index {{math|''n''}}. Doing so will show that it is an ] for even values of {{math|''k''}}, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that {{math|''B''<sub>2''k'' + 1 − ''m''</sub>}} is 0 for {{math|''m''}} even and {{math|2''k'' + 1 − ''m'' > 1}}; and that the term for {{math|''B''<sub>1</sub>}} is cancelled by the subtraction. The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for ''n'' > 1).


From the von Staudt-Clausen theorem it is known that for odd ''n''&nbsp;&gt;&nbsp;1 the number 2''B''<sub>''n''</sub> is an integer. This seems trivial if one knows beforehand that in this case ''B''<sub>''n''</sub>&nbsp;=&nbsp;0. However, by applying Worpitzky's representation one gets From the von Staudt–Clausen theorem it is known that for odd {{math|''n'' > 1}} the number {{math|2''B''<sub>''n''</sub>}} is an integer. This seems trivial if one knows beforehand that the integer in question is zero. However, by applying Worpitzky's representation one gets


: <math> 2B_n =\sum_{m=0}^n \left(-1\right)^m \frac{2}{m+1}m! : <math> 2B_n =\sum_{m=0}^n (-1)^m \frac{2}{m+1}m! \left\{{n+1\atop m+1} \right\} = 0\quad(n>1 \text{ is odd})</math>
\left\{\begin{matrix} n+1 \\ m+1 \end{matrix}\right\}
=0\quad\left(n>1\ \text{odd}\right) </math>


as a ''sum of integers'', which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let ''S''<sub>''n'',''m''</sub> be the number of surjective maps from {1,&nbsp;2,&nbsp;...,&nbsp;''n''} to {1,&nbsp;2,&nbsp;...,&nbsp;''m''}, then <math>S_{n,m}=m! \left\{\begin{matrix} n \\ m \end{matrix}\right\} </math>. The last equation can only hold if as a ''sum of integers'', which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let {{math|''S''<sub>''n'',''m''</sub>}} be the number of surjective maps from {{math|1={1, 2, ..., ''n''}}} to {{math|1={1, 2, ..., ''m''}}}, then {{math|''S''<sub>''n'',''m''</sub> {{=}} ''m''!<big><big>{</big></big>{{su|p=''n''|b=''m''|a=c}}<big><big>}</big></big>}}. The last equation can only hold if


: <math> \sum_{m=1,3,5,\ldots\leq n}\frac{2}{m^{2}}S_{n,m}=\sum_{m=2,4,6,\ldots\leq n} \frac{2}{m^{2}} S_{n,m} \quad \left(n>2 \text{ even}\right) \ . </math> : <math> \sum_{\text{odd }m=1}^{n-1} \frac 2 {m^2}S_{n,m}=\sum_{\text{even } m=2}^n \frac{2}{m^2} S_{n,m} \quad (n>2 \text{ is even}). </math>


This equation can be proved by induction. The first two examples of this equation are This equation can be proved by induction. The first two examples of this equation are


:{{math|1=''n'' = 4: 2 + 8 = 7 + 3}},
''n''&nbsp;=&nbsp;4&nbsp;:&nbsp; 2&nbsp;+&nbsp;8&nbsp;=&nbsp;7&nbsp;+&nbsp;3,&nbsp; ''n''&nbsp;=&nbsp;6:&nbsp; 2&nbsp;+&nbsp;120&nbsp;+&nbsp;144&nbsp;=&nbsp;31&nbsp;+&nbsp;195&nbsp;+&nbsp;40.
:{{math|1=''n'' = 6: 2 + 120 + 144 = 31 + 195 + 40}}.


Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers. Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.


=== A restatement of the Riemann hypothesis ===
===''p''-adic continuity===
The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the ] (RH) which uses only the Bernoulli numbers. In fact ] proved that the RH is equivalent to the following assertion:{{r|Riesz1916}}


:For every {{math|''ε'' > {{sfrac|1|4}}}} there exists a constant {{math|''C''<sub>''ε''</sub> > 0}} (depending on {{math|''ε''}}) such that {{math|{{abs|''R''(''x'')}} < ''C''<sub>''ε''</sub>''x''<sup>''ε''</sup>}} as {{math|''x'' → ∞}}.
An especially important congruence property of the Bernoulli numbers can be characterized as a p-adic continuity property. If ''b'', ''m'' and ''n'' are positive integers such that ''m'' and ''n'' are not divisible by ''p''&nbsp;−&nbsp;1 and <math>m \equiv n\, \bmod\,p^{b-1}(p-1)</math>, then


Here {{math|''R''(''x'')}} is the ]
:<math>(1-p^{m-1}){B_m \over m} \equiv (1-p^{n-1}){B_n \over n} \,\bmod\, p^b.</math>


: <math> R(x) = 2 \sum_{k=1}^\infty
Since <math>B_n = -n\zeta(1-n)</math>, this can also be written
\frac{k^{\overline{k}} x^{k}}{(2\pi)^{2k}\left(\frac{B_{2k}}{2k}\right)}
= 2\sum_{k=1}^\infty \frac{k^{\overline{k}}x^k}{(2\pi)^{2k}\beta_{2k}}. </math>


{{math|''n''<sup>{{overline|''k''}}</sup>}} denotes the ] in the notation of ]. The numbers {{math|''β''<sub>''n''</sub> {{=}} {{sfrac|''B''<sub>''n''</sub>|''n''}}}} occur frequently in the study of the zeta function and are significant because {{math|''β''<sub>''n''</sub>}} is a {{math|''p''}}-integer for primes {{math|''p''}} where {{math|''p'' − 1}} does not divide {{math|''n''}}. The {{math|''β''<sub>''n''</sub>}} are called ''divided Bernoulli numbers''.
:<math>(1-p^{-u})\zeta(u) \equiv (1-p^{-v})\zeta(v)\, \bmod \,p^b\,,</math>


==<span id="Generalized Bernoulli numbers"></span>Generalized Bernoulli numbers==
where ''u'' = 1 − ''m'' and ''v'' = 1 − ''n'', so that ''u'' and ''v'' are nonpositive and not congruent to 1 mod ''p''&nbsp;−&nbsp;1. This tells us that the Riemann zeta function, with 1&nbsp;−&nbsp;''p''<sup>−''s''</sup> taken out of the Euler product formula, is continuous in the ]s on odd negative integers congruent mod ''p''&nbsp;−&nbsp;1 to a particular <math>a \not\equiv 1\, \bmod\, p-1</math>, and so can be extended to a continuous function <math>\zeta_p(s)</math> for all ''p''-adic integers <math>\Bbb{Z}_p,\,</math> the '''''p''-adic Zeta function'''.
The '''generalized Bernoulli numbers''' are certain ]s, defined similarly to the Bernoulli numbers, that are related to ] of ] in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.


Let {{mvar|χ}} be a ] modulo {{mvar|f}}. The generalized Bernoulli numbers attached to {{mvar|χ}} are defined by
=== Bernoulli numbers and the Riemann hypothesis ===


: <math>\sum_{a=1}^f \chi(a) \frac{te^{at}}{e^{ft}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}.</math>
The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the ] (RH) which uses only the Bernoulli number. In fact in 1916 M. Riesz proved that the RH is equivalent to the following assertion:


Apart from the exceptional {{math|''B''<sub>1,1</sub> {{=}} {{sfrac|1|2}}}}, we have, for any Dirichlet character {{mvar|χ}}, that {{math|''B''<sub>''k'',''χ''</sub> {{=}} 0}} if {{math|''χ''(−1) ≠ (−1)<sup>''k''</sup>}}.
For every ''ε''&nbsp;>&nbsp;1/4 there exists a constant ''C''<sub>''ε''</sub>&nbsp;>&nbsp;0
(depending on ''ε'') such that |''R''(''x'')|&nbsp;<&nbsp;''C''<sub>ε</sub>&nbsp;''x''<sup>ε</sup> as ''x''&nbsp;→&nbsp;∞. Here ''R''(''x'') is the ]


Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers {{math|''k'' ≥ 1}}:
: <math> R(x) = 2 \sum_{k=1}^{\infty}
\frac{k^{\overline{k}} x^{k}}{(2\pi)^{2k}\left(B_{2k}/(2k)\right)}
= 2\sum_{k=1}^{\infty}\frac{k^{\overline{k}}x^{k}}{(2\pi)^{2k}\beta_{2k}}. \ </math>


: <math>L(1-k,\chi)=-\frac{B_{k,\chi}}k,</math>
<math>n^{\overline{k}}</math> denotes the ] in the notation of ]. The number ''β''<sub>''n''</sub> = ''B''<sub>''n''</sub>/''n'' occur frequently in the study of the zeta function and are significant because ''β''<sub>''n''</sub> is a ''p''-integer for primes ''p'' where ''p''&nbsp;−&nbsp;1 does not divide ''n''. The ''β''<sub>''n''</sub> are called ''divided Bernoulli number''.


where {{math|''L''(''s'',''χ'')}} is the Dirichlet {{mvar|L}}-function of {{mvar|χ}}.{{r|Neukirch1999_VII2}}
=== Ramanujan's congruences ===


===Eisenstein–Kronecker number===
The following relations, due to ], provide a more efficient method for calculating Bernoulli numbers:
{{main|Eisenstein–Kronecker number}}
]s are an analogue of the generalized Bernoulli numbers for ]s.{{r|Charollois-Sczech|BK}} They are related to critical ''L''-values of ]s.{{r|BK}}


==Appendix==
:<math>m\equiv 0\,\bmod\,6\qquad {{m+3}\choose{m}}B_m={{m+3}\over3}-\sum_{j=1}^{m/6}{m+3\choose{m-6j}}B_{m-6j}</math>
=== Assorted identities ===
{{unordered list
|1 = ] gives a compact form of Bernoulli's formula by using an abstract symbol {{math|'''B'''}}:


:<math>m\equiv 2\,\bmod\,6\qquad {{m+3}\choose{m}}B_m={{m+3}\over3}-\sum_{j=1}^{(m-2)/6}{m+3\choose{m-6j}}B_{m-6j}</math> : <math>S_m(n) = \frac 1 {m+1} ((\mathbf{B} + n)^{m+1} - B_{m+1}) </math>


where the symbol {{math|'''B'''<sup>''k''</sup>}} that appears during binomial expansion of the parenthesized term is to be replaced by the Bernoulli number {{math|''B<sub>k</sub>''}} (and {{math|''B''<sub>1</sub> {{=}} +{{sfrac|1|2}}}}). More suggestively and mnemonically, this may be written as a definite integral:
:<math>m\equiv 4\,\bmod\, 6\qquad{{m+3}\choose{m}}B_m=-{{m+3}\over6}-\sum_{j=1}^{(m-4)/6}{m+3\choose{m-6j}}B_{m-6j}.</math>


:<math>S_m(n) = \int_0^n (\mathbf{B}+x)^m\,dx </math>
== Use of Bernoulli numbers in topology ==


Many other Bernoulli identities can be written compactly with this symbol, e.g.
The ] for the order of the cyclic group of diffeomorphism classes of ] which bound ]s for <math>n \ge 2</math> involves Bernoulli numbers; if ''B''<sub>(''n'')</sub> is the numerator of ''B''<sub>4''n''</sub>/''n'', then


:<math>2^{2n-2}(1-2^{2n-1})B_{(n)}</math> :<math> (1-2\mathbf{B})^m = (2-2^m) B_m </math>


|2 = Let {{math|''n''}} be non-negative and even
is the number of such exotic spheres. (The formula in the topological literature differs because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)


:<math> \zeta(n) = \frac{(-1)^{\frac{n}{2} - 1} B_n (2\pi)^n}{2(n!)}</math>
== Assorted identities ==
<ul>
<li>


|3 = The {{math|''n''}}th ] of the ] ] on the interval is {{math|{{sfrac|''B''<sub>''n''</sub>|''n''}}}}.
Let ''n'' be non-negative and even


|4 = Let {{math|''n''? {{=}} {{sfrac|1|''n''!}}}} and {{math|''n'' ≥ 1}}. Then {{math|''B''<sub>''n''</sub>}} is the following {{math|(''n'' + 1) × (''n'' + 1)}} determinant:{{r|Malenfant2011}}
:<math> \zeta(n) = \frac{\left(-1\right)^{\frac{n}{2}-1}B_n\left(2\pi\right)^n}{2n!}</math>


</li> : <math>
\begin{align}
<li>
B_n & = n! \begin{vmatrix}
The ''n''th ] of the ] ] on the interval is ''B''<sub>''n''</sub>/''n''.
1 & 0 & \cdots & 0 & 1 \\
</li>
2? & 1 & \cdots & 0 & 0 \\
<li>
\vdots & \vdots & & \vdots & \vdots \\
Let ''n''¡ = 1/''n''! and ''n'' ≥ 1.
n? & (n-1)? & \cdots & 1 & 0 \\
Then ''B''<sub>''n''</sub> is ''n''! times the
(n+1)? & n? & \cdots & 2? & 0
determinant of the following matrix.
\end{vmatrix} \\
& = n! \begin{vmatrix}
1 & 0 & \cdots & 0 & 1 \\
\frac{1}{2!} & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & & \vdots & \vdots \\
\frac{1}{n!} & \frac{1}{(n-1)!} & \cdots & 1 & 0 \\
\frac{1}{(n+1)!} & \frac{1}{n!} & \cdots & \frac{1}{2!} & 0
\end{vmatrix}
\end{align}
</math>


Thus the determinant is {{math|''σ''<sub>''n''</sub>(1)}}, the ] at {{math|''x'' {{=}} 1}}.
<table>
<tr>
<td>''B''<sub>''n''</sub>&nbsp;=&nbsp;</td>
<td><table cellspacing="4" cellpadding="4" style="text-align: center" >
<tr>
<td> 1 <br > ───<br >''n''¡</td>
</tr>
</table>
</td>
<td>
<table style="text-align: center; border-left: 2px solid #000000;
border-right: 2px solid #000000" cellspacing="4" cellpadding="4">
<tr>
<td valign="top">2¡</td>
<td valign="top">1¡</td>
<td valign="top">0</td>
<td valign="top">0</td>
<td valign="top">0</td>
<td valign="top">...</td>
<td valign="top">0</td>
</tr>
<tr>
<td valign="top">3¡</td>
<td valign="top">2¡</td>
<td valign="top">1¡</td>
<td valign="top">0</td>
<td valign="top">0</td>
<td valign="top">...</td>
<td valign="top">0</td>
</tr>
<tr>
<td valign="top">4¡</td>
<td valign="top">3¡</td>
<td valign="top">2¡</td>
<td valign="top">1¡</td>
<td valign="top">0</td>
<td valign="top">...</td>
<td valign="top">0</td>
</tr>
<tr>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">...</td>
</tr>
<tr>
<td valign="top">(''n''&nbsp;−&nbsp;2)¡</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">3¡</td>
<td valign="top">2¡</td>
<td valign="top">1¡</td>
<td valign="top">0</td>
</tr>
<tr>
<td valign="top">(''n''&nbsp;−&nbsp;1)¡</td>
<td valign="top">(''n''&nbsp;−&nbsp;2)¡</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">3¡</td>
<td valign="top">2¡</td>
<td valign="top">1¡</td>
</tr>
<tr>
<td valign="top">''n''¡</td>
<td valign="top">(''n''&nbsp;−&nbsp;1)¡</td>
<td valign="top">(''n''&nbsp;−&nbsp;2)¡</td>
<td valign="top">...</td>
<td valign="top">...</td>
<td valign="top">3¡</sub></td>
<td valign="top">2¡</sub></td>
</tr>
</table>
</td>
</tr>
</table>


|5 = For even-numbered Bernoulli numbers, {{math|''B''<sub>2''p''</sub>}} is given by the {{math|(''p'' + 1) × (''p'' + 1)}} determinant::{{r|Malenfant2011}}
Thus the determinant is σ<sub>''n''</sub>(1), the Stirling polynomial at ''x'' = 1.


:<math> B_{2p} = -\frac{(2p)!}{2^{2p} - 2} \begin{vmatrix}
</li>
1 & 0 & 0 & \cdots & 0 & 1 \\
<li>
\frac{1}{3!} & 1 & 0 & \cdots & 0 & 0 \\
Let ''n'' ≥ 1.
\frac{1}{5!} & \frac{1}{3!} & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots& \vdots \\
\frac{1}{(2p+1)!} & \frac{1}{(2p-1)!} & \frac{1}{(2p-3)!} &\cdots & \frac{1}{3!} & 0
\end{vmatrix}</math>


|6 = Let {{math|''n'' ≥ 1}}. Then (])<ref>Euler, E41, Inventio summae cuiusque seriei ex dato termino generali </ref>
: <math> \frac{1}{n} \sum_{k=1}^{n}\binom{n}{k}B_{k}B_{n-k}+B_{n-1}=-B_{n} \quad \text{(L. Euler)} </math>
</li>
<li>


: <math> \frac{1}{n} \sum_{k=1}^n \binom{n}{k}B_k B_{n-k}+B_{n-1}=-B_n </math>
Let ''n'' ≥ 1. Then {{harv|von Ettingshausen|1827}}


|7 = Let {{math|''n'' ≥ 1}}. Then{{r|vonEttingshausen1827}}
: <math> \sum_{k=0}^{n}\binom{n+1}{k}(n+k+1)B_{n+k}=0 </math>


: <math> \sum_{k=0}^n \binom{n+1}k (n+k+1)B_{n+k}=0 </math>
</li>
<li>


Let ''n'' ≥ 1 and ''m'' ≥ 1. Then {{harv|Carlitz|1968}} |8 = Let {{math|''n'' ≥ 0}}. Then (] 1883)


: <math> (-1)^{m}\sum_{r=0}^{m}\binom{m}{r}B_{n+r}=(-1)^{n}\sum_{s=0}^{n}\binom{n}{s}B_{m+s} </math> : <math> B_n = - \sum_{k=1}^{n+1} \frac{(-1)^k}{k} \binom{n+1}{k} \sum_{j=1}^k j^n </math>


|9 = Let {{math|''n'' ≥ 1}} and {{math|''m'' ≥ 1}}. Then{{r|Carlitz1968}}
</li><li>


: <math> (-1)^m \sum_{r=0}^m \binom{m}{r} B_{n+r}=(-1)^n \sum_{s=0}^n \binom{n}{s} B_{m+s} </math>
Let ''n'' ≥ 4 and


|10 = Let {{math|''n'' ≥ 4}} and
: <math> H_{n}=\sum_{1\leq k\leq n}k^{-1} </math>


: <math> H_n=\sum_{k=1}^n k^{-1} </math>
the ]. Then


the ]. Then (H. Miki 1978)
: <math> \frac{n}{2}\sum_{k=2}^{n-2}\frac{B_{n-k}}{n-k}\frac{B_{k}}{k}-\sum_{k=2}
^{n-2}\binom{n}{k}\frac{B_{n-k}}{n-k}B_{k}=H_{n}B_{n}\qquad\text{(H. Miki, 1978)} </math>


: <math> \frac{n}{2}\sum_{k=2}^{n-2}\frac{B_{n-k}}{n-k}\frac{B_k}{k} - \sum_{k=2}^{n-2} \binom{n}{k}\frac{B_{n-k}}{n-k} B_k =H_n B_n</math>
</li><li>


Let ''n'' ≥ 4. Yuri ] found (1997) |11 = Let {{math|''n'' ≥ 4}}. ] found (1997)


: <math> (n+2)\sum_{k=2}^{n-2}B_{k}B_{n-k}-2\sum_{l=2}^{n-2}\binom{n+2}{l}B_{l} B_{n-l}=n(n+1)B_{n} </math> : <math> (n+2)\sum_{k=2}^{n-2}B_k B_{n-k}-2\sum_{l=2}^{n-2}\binom{n+2}{l} B_l B_{n-l}=n(n+1)B_n </math>


|12 = ''Faber–]–]–Gessel identity'': for {{math|''n'' ≥ 1}},
</li><li>

Let ''n'' ≥ 1


: <math> \frac{n}{2}\left(B_{n-1}(x)+\sum_{k=1}^{n-1}\frac{B_{k}(x)}{k} : <math> \frac{n}{2}\left(B_{n-1}(x)+\sum_{k=1}^{n-1}\frac{B_{k}(x)}{k}
\frac{B_{n-k}(x)}{n-k}\right) -\sum_{k=0}^{n-1}\binom{n}{k}\frac{B_{n-k}} \frac{B_{n-k}(x)}{n-k}\right) -\sum_{k=0}^{n-1}\binom{n}{k}\frac{B_{n-k}}
{n-k}B_{k}(x)=H_{n-1}B_{n}(x) </math> {n-k} B_k(x) =H_{n-1}B_n(x).</math>


This is an identity by ''Faber-Pandharipande-]-Gessel.'' Choose ''x'' = 0 or ''x'' = 1 to get a Bernoulli number identity according to your favourite convention. Choosing {{math|''x'' {{=}} 0}} or {{math|''x'' {{=}} 1}} results in the Bernoulli number identity in one or another convention.
</li>


|13 = The next formula is true for {{math|''n'' ≥ 0}} if {{math|''B''<sub>1</sub> {{=}} ''B''<sub>1</sub>(1) {{=}} {{sfrac|1|2}}}}, but only for {{math|''n'' ≥ 1}} if {{math|''B''<sub>1</sub> {{=}} ''B''<sub>1</sub>(0) {{=}} −{{sfrac|1|2}}}}.
<li>
The next formula is true for ''n'' ≥ 0 if ''B''<sub>1</sub> = ''B''<sub>1</sub>(1) = 1/2, but only for ''n'' ≥ 1 if ''B''<sub>1</sub> = ''B''<sub>1</sub>(0) = -1/2.


:<math> \sum_{k=0}^{n}\binom{n}{k} \frac{B_{k}}{n-k+2} = \frac{B_{n+1}}{n+1} </math> :<math> \sum_{k=0}^n \binom{n}{k} \frac{B_k}{n-k+2} = \frac{B_{n+1}}{n+1} </math>


|14 = Let {{math|''n'' ≥ 0}}. Then
</li>
<li>
Let ''n'' ≥ 0 and = 1 if ''b'' is true, 0 otherwise.


:<math> -1 + \sum_{k=0}^{n}\binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_{k}(1) = 2^{n} </math> :<math> -1 + \sum_{k=0}^n \binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_k(1) = 2^n </math>


and and


:<math> -1 + \sum_{k=0}^{n}\binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_{k}(0) = </math> :<math> -1 + \sum_{k=0}^n \binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_{k}(0) = \delta_{n,0} </math>


|15 = A reciprocity relation of M.&nbsp;B.&nbsp;Gelfand:{{r|AgohDilcher2008}}
</li>

</ul>
: <math> (-1)^{m+1} \sum_{j=0}^k \binom{k}{j} \frac{B_{m+1+j}}{m+1+j} + (-1)^{k+1} \sum_{j=0}^m \binom{m}{j}\frac{B_{k+1+j}}{k+1+j} = \frac{k!m!}{(k+m+1)!} </math>

}}


== See also == == See also ==
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* ]
* ] * ]
* ] * ]
* ] * ]
* ] * ]
* ] * ]
* ]
* ]
* ]
* ]
* ] * ]
* ]
* ]
{{div col end}}


== Notes == == Notes ==
{{Notelist}}
<references group="~" />


== References == == References ==
{{reflist|refs=
{{Reflist}}


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==External links== ==External links==
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* {{citation |author=P. Luschny |title= The Computation of Irregular Primes |url=http://www.luschny.de/math/primes/irregular.html}}
* {{citation|title=Bernoulli numbers. Bibliography (1713-1990)|first1=K.|last1=Dilcher|first2=L.|last2=Skula|first3=I. Sh.|last3=Slavutskii|journal=Queen's Papers in Pure and Applied Mathematics|number=87|publication-place=Kingston, Ontario|year=1991|url=http://www.mscs.dal.ca/~dilcher/bernoulli.html}}
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* {{citation |author=Gottfried Helms |title= Bernoullinumbers in context of Pascal-(Binomial)matrix |url=http://go.helms-net.de/math/pascal/bernoulli_en.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://go.helms-net.de/math/pascal/bernoulli_en.pdf |archive-date=2022-10-09 |url-status=live}}
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* {{citation |author=Gottfried Helms |title=summing of like powers in context with Pascal-/Bernoulli-matrix |url=http://go.helms-net.de/math/binomial/04_3_SummingOfLikePowers.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://go.helms-net.de/math/binomial/04_3_SummingOfLikePowers.pdf |archive-date=2022-10-09 |url-status=live}}
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* * {{citation |author=Gottfried Helms |title=Some special properties, sums of Bernoulli-and related numbers |url=http://go.helms-net.de/math/binomial/02_2_GeneralizedBernoulliRecursion.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://go.helms-net.de/math/binomial/02_2_GeneralizedBernoulliRecursion.pdf |archive-date=2022-10-09 |url-status=live}}

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Latest revision as of 00:11, 17 December 2024

Rational number sequence

Bernoulli numbers B
n
n fraction decimal
0 1 +1.000000000
1 ±⁠1/2⁠ ±0.500000000
2 ⁠1/6⁠ +0.166666666
3 0 +0.000000000
4 −⁠1/30⁠ −0.033333333
5 0 +0.000000000
6 ⁠1/42⁠ +0.023809523
7 0 +0.000000000
8 −⁠1/30⁠ −0.033333333
9 0 +0.000000000
10 ⁠5/66⁠ +0.075757575
11 0 +0.000000000
12 −⁠691/2730⁠ −0.253113553
13 0 +0.000000000
14 ⁠7/6⁠ +1.166666666
15 0 +0.000000000
16 −⁠3617/510⁠ −7.092156862
17 0 +0.000000000
18 ⁠43867/798⁠ +54.97117794
19 0 +0.000000000
20 −⁠174611/330⁠ −529.1242424

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B n − {\displaystyle B_{n}^{-{}}} and B n + {\displaystyle B_{n}^{+{}}} ; they differ only for n = 1, where B 1 − = − 1 / 2 {\displaystyle B_{1}^{-{}}=-1/2} and B 1 + = + 1 / 2 {\displaystyle B_{1}^{+{}}=+1/2} . For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with B n − = B n ( 0 ) {\displaystyle B_{n}^{-{}}=B_{n}(0)} and B n + = B n ( 1 ) {\displaystyle B_{n}^{+}=B_{n}(1)} .

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine; it is disputed whether Lovelace or Babbage developed the algorithm. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

Notation

The superscript ± used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the n = 1 term is affected:

  • B
    n with B
    1 = −⁠1/2⁠ (OEIS: A027641 / OEIS: A027642) is the sign convention prescribed by NIST and most modern textbooks.
  • B
    n with B
    1 = +⁠1/2⁠ (OEIS: A164555 / OEIS: A027642) was used in the older literature, and (since 2022) by Donald Knuth following Peter Luschny's "Bernoulli Manifesto".

In the formulas below, one can switch from one sign convention to the other with the relation B n + = ( − 1 ) n B n − {\displaystyle B_{n}^{+}=(-1)^{n}B_{n}^{-}} , or for integer n = 2 or greater, simply ignore it.

Since Bn = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, a few authors write "Bn" instead of B2n . This article does not follow that notation.

History

Early history

The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.

A page from Seki Takakazu's Katsuyō Sanpō (1712), tabulating binomial coefficients and Bernoulli numbers

Methods to calculate the sum of the first n positive integers, the sum of the squares and of the cubes of the first n positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras (c. 572–497 BCE, Greece), Archimedes (287–212 BCE, Italy), Aryabhata (b. 476, India), Abu Bakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham (965–1039, Iraq).

During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West Thomas Harriot (1560–1621) of England, Johann Faulhaber (1580–1635) of Germany, Pierre de Fermat (1601–1665) and fellow French mathematician Blaise Pascal (1623–1662) all played important roles.

Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae, far higher than anyone before him, but he did not give a general formula.

Blaise Pascal in 1654 proved Pascal's identity relating (n+1) to the sums of the pth powers of the first n positive integers for p = 0, 1, 2, ..., k.

The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1, B2,... which provide a uniform formula for all sums of powers.

The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the cth powers for any positive integer c can be seen from his comment. He wrote:

"With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500."

Bernoulli's result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712. However, Seki did not present his method as a formula based on a sequence of constants.

Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of Abraham de Moivre.

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834. Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on):

"Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0, B1, B2, ... would provide a uniform
∑ n m = 1 m + 1 ( B 0 n m + 1 − ( m + 1 1 ) B 1 n m + ( m + 1 2 ) B 2 n m − 1 − ⋯ + ( − 1 ) m ( m + 1 m ) B m n ) {\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}-{\binom {m+1}{1}}B_{1}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}-\cdots +(-1)^{m}{\binom {m+1}{m}}B_{m}n\right)}
for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for Σ n from polynomials in N to polynomials in n."

In the above Knuth meant B 1 − {\displaystyle B_{1}^{-}} ; instead using B 1 + {\displaystyle B_{1}^{+}} the formula avoids subtraction:

∑ n m = 1 m + 1 ( B 0 n m + 1 + ( m + 1 1 ) B 1 + n m + ( m + 1 2 ) B 2 n m − 1 + ⋯ + ( m + 1 m ) B m n ) . {\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}+{\binom {m+1}{1}}B_{1}^{+}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}+\cdots +{\binom {m+1}{m}}B_{m}n\right).}

Reconstruction of "Summae Potestatum"

Jakob Bernoulli's "Summae Potestatum", 1713

The Bernoulli numbers OEIS: A164555(n)/OEIS: A027642(n) were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted A, B, C and D by Bernoulli are mapped to the notation which is now prevalent as A = B2, B = B4, C = B6, D = B8. The expression c·c−1·c−2·c−3 means c·(c−1)·(c−2)·(c−3) – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers c. The factorial notation k! as a shortcut for 1 × 2 × ... × k was not introduced until 100 years later. The integral symbol on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter S for "summa" (sum). The letter n on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as 1, 2, ..., n. Putting things together, for positive c, today a mathematician is likely to write Bernoulli's formula as:

∑ k = 1 n k c = n c + 1 c + 1 + 1 2 n c + ∑ k = 2 c B k k ! c k − 1 _ n c − k + 1 . {\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}

This formula suggests setting B1 = ⁠1/2⁠ when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the falling factorial c has for k = 0 the value ⁠1/c + 1⁠. Thus Bernoulli's formula can be written

∑ k = 1 n k c = ∑ k = 0 c B k k ! c k − 1 _ n c − k + 1 {\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}

if B1 = 1/2, recapturing the value Bernoulli gave to the coefficient at that position.

The formula for ∑ k = 1 n k 9 {\displaystyle \textstyle \sum _{k=1}^{n}k^{9}} in the first half of the quotation by Bernoulli above contains an error at the last term; it should be − 3 20 n 2 {\displaystyle -{\tfrac {3}{20}}n^{2}} instead of − 1 12 n 2 {\displaystyle -{\tfrac {1}{12}}n^{2}} .

Definitions

Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned:

  • a recursive equation,
  • an explicit formula,
  • a generating function,
  • an integral expression.

For the proof of the equivalence of the four approaches.

Recursive definition

The Bernoulli numbers obey the sum formulas

∑ k = 0 m ( m + 1 k ) B k − = δ m , 0 ∑ k = 0 m ( m + 1 k ) B k + = m + 1 {\displaystyle {\begin{aligned}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{-{}}&=\delta _{m,0}\\\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+{}}&=m+1\end{aligned}}}

where m = 0 , 1 , 2... {\displaystyle m=0,1,2...} and δ denotes the Kronecker delta.

The first of these is sometimes written as the formula (for m > 1) ( B + 1 ) m − B m = 0 , {\displaystyle (B+1)^{m}-B_{m}=0,} where the power is expanded formally using the binomial theorem and B k {\displaystyle B^{k}} is replaced by B k {\displaystyle B_{k}} .

Solving for B m ∓ {\displaystyle B_{m}^{\mp {}}} gives the recursive formulas

B m − = δ m , 0 − ∑ k = 0 m − 1 ( m k ) B k − m − k + 1 B m + = 1 − ∑ k = 0 m − 1 ( m k ) B k + m − k + 1 . {\displaystyle {\begin{aligned}B_{m}^{-{}}&=\delta _{m,0}-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{-{}}}{m-k+1}}\\B_{m}^{+}&=1-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{+}}{m-k+1}}.\end{aligned}}}

Explicit definition

In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers, usually giving some reference in the older literature. One of them is (for m ≥ 1 {\displaystyle m\geq 1} ):

B m − = ∑ k = 0 m 1 k + 1 ∑ j = 0 k ( k j ) ( − 1 ) j j m B m + = ∑ k = 0 m 1 k + 1 ∑ j = 0 k ( k j ) ( − 1 ) j ( j + 1 ) m . {\displaystyle {\begin{aligned}B_{m}^{-}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}j^{m}\\B_{m}^{+}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}(j+1)^{m}.\end{aligned}}}

Generating function

The exponential generating functions are

t e t − 1 = t 2 ( coth ⁡ t 2 − 1 ) = ∑ m = 0 ∞ B m − t m m ! t e t e t − 1 = t 1 − e − t = t 2 ( coth ⁡ t 2 + 1 ) = ∑ m = 0 ∞ B m + t m m ! . {\displaystyle {\begin{alignedat}{3}{\frac {t}{e^{t}-1}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}-1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{-{}}t^{m}}{m!}}\\{\frac {te^{t}}{e^{t}-1}}={\frac {t}{1-e^{-t}}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}+1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{+}t^{m}}{m!}}.\end{alignedat}}}

where the substitution is t → − t {\displaystyle t\to -t} . The two generating functions only differ by t.

Proof

If we let F ( t ) = ∑ i = 1 ∞ f i t i {\displaystyle F(t)=\sum _{i=1}^{\infty }f_{i}t^{i}} and G ( t ) = 1 / ( 1 + F ( t ) ) = ∑ i = 0 ∞ g i t i {\displaystyle G(t)=1/(1+F(t))=\sum _{i=0}^{\infty }g_{i}t^{i}} then

G ( t ) = 1 − F ( t ) G ( t ) . {\displaystyle G(t)=1-F(t)G(t).}

Then g 0 = 1 {\displaystyle g_{0}=1} and for m > 0 {\displaystyle m>0} the m term in the series for G ( t ) {\displaystyle G(t)} is:

g m t i = − ∑ j = 0 m − 1 f m − j g j t m {\displaystyle g_{m}t^{i}=-\sum _{j=0}^{m-1}f_{m-j}g_{j}t^{m}}

If

F ( t ) = e t − 1 t − 1 = ∑ i = 1 ∞ t i ( i + 1 ) ! {\displaystyle F(t)={\frac {e^{t}-1}{t}}-1=\sum _{i=1}^{\infty }{\frac {t^{i}}{(i+1)!}}}

then we find that

G ( t ) = t / ( e t − 1 ) {\displaystyle G(t)=t/(e^{t}-1)}
m ! g m = − ∑ j = 0 m − 1 m ! j ! j ! g j ( m − j + 1 ) ! = − 1 m + 1 ∑ j = 0 m − 1 ( m + 1 j ) j ! g j {\displaystyle {\begin{aligned}m!g_{m}&=-\sum _{j=0}^{m-1}{\frac {m!}{j!}}{\frac {j!g_{j}}{(m-j+1)!}}\\&=-{\frac {1}{m+1}}\sum _{j=0}^{m-1}{\binom {m+1}{j}}j!g_{j}\\\end{aligned}}}

showing that the values of i ! g i {\displaystyle i!g_{i}} obey the recursive formula for the Bernoulli numbers B i − {\displaystyle B_{i}^{-}} .

The (ordinary) generating function

z − 1 ψ 1 ( z − 1 ) = ∑ m = 0 ∞ B m + z m {\displaystyle z^{-1}\psi _{1}(z^{-1})=\sum _{m=0}^{\infty }B_{m}^{+}z^{m}}

is an asymptotic series. It contains the trigamma function ψ1.

Integral Expression

From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers:

B 2 n = 4 n ( − 1 ) n + 1 ∫ 0 ∞ t 2 n − 1 e 2 π t − 1 d t {\displaystyle B_{2n}=4n(-1)^{n+1}\int _{0}^{\infty }{\frac {t^{2n-1}}{e^{2\pi t}-1}}\mathrm {d} t}

Bernoulli numbers and the Riemann zeta function

The Bernoulli numbers as given by the Riemann zeta function.

The Bernoulli numbers can be expressed in terms of the Riemann zeta function:

B
n = −n ζ(1 − n)           for n ≥ 1 .

Here the argument of the zeta function is 0 or negative. As ζ ( k ) {\displaystyle \zeta (k)} is zero for negative even integers (the trivial zeroes), if n>1 is odd, ζ ( 1 − n ) {\displaystyle \zeta (1-n)} is zero.

By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:

B 2 n = ( − 1 ) n + 1 2 ( 2 n ) ! ( 2 π ) 2 n ζ ( 2 n ) {\displaystyle B_{2n}={\frac {(-1)^{n+1}2(2n)!}{(2\pi )^{2n}}}\zeta (2n)\quad } for n ≥ 1 .

Now the argument of the zeta function is positive.

It then follows from ζ → 1 (n → ∞) and Stirling's formula that

| B 2 n | ∼ 4 π n ( n π e ) 2 n {\displaystyle |B_{2n}|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}\quad } for n → ∞ .

Efficient computation of Bernoulli numbers

In some applications it is useful to be able to compute the Bernoulli numbers B0 through Bp − 3 modulo p, where p is a prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p arithmetic operations would be required. Fortunately, faster methods have been developed which require only O(p (log p)) operations (see big O notation).

David Harvey describes an algorithm for computing Bernoulli numbers by computing Bn modulo p for many small primes p, and then reconstructing Bn via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is O(n log(n)) and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed Bn for n = 10. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner computed Bn to full precision for n = 10 in December 2002 and Oleksandr Pavlyk for n = 10 with Mathematica in April 2008.

Computer Year n Digits*
J. Bernoulli ~1689 10 1
L. Euler 1748 30 8
J. C. Adams 1878 62 36
D. E. Knuth, T. J. Buckholtz 1967 1672 3330
G. Fee, S. Plouffe 1996 10000 27677
G. Fee, S. Plouffe 1996 100000 376755
B. C. Kellner 2002 1000000 4767529
O. Pavlyk 2008 10000000 57675260
D. Harvey 2008 100000000 676752569
* Digits is to be understood as the exponent of 10 when Bn is written as a real number in normalized scientific notation.

Applications of the Bernoulli numbers

Asymptotic analysis

Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula. Assuming that f is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as

∑ k = a b − 1 f ( k ) = ∫ a b f ( x ) d x + ∑ k = 1 m B k − k ! ( f ( k − 1 ) ( b ) − f ( k − 1 ) ( a ) ) + R − ( f , m ) . {\displaystyle \sum _{k=a}^{b-1}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{-}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{-}(f,m).}

This formulation assumes the convention B
1 = −⁠1/2⁠. Using the convention B
1 = +⁠1/2⁠ the formula becomes

∑ k = a + 1 b f ( k ) = ∫ a b f ( x ) d x + ∑ k = 1 m B k + k ! ( f ( k − 1 ) ( b ) − f ( k − 1 ) ( a ) ) + R + ( f , m ) . {\displaystyle \sum _{k=a+1}^{b}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{+}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{+}(f,m).}

Here f ( 0 ) = f {\displaystyle f^{(0)}=f} (i.e. the zeroth-order derivative of f {\displaystyle f} is just f {\displaystyle f} ). Moreover, let f ( − 1 ) {\displaystyle f^{(-1)}} denote an antiderivative of f {\displaystyle f} . By the fundamental theorem of calculus,

∫ a b f ( x ) d x = f ( − 1 ) ( b ) − f ( − 1 ) ( a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=f^{(-1)}(b)-f^{(-1)}(a).}

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

∑ k = a + 1 b f ( k ) = ∑ k = 0 m B k k ! ( f ( k − 1 ) ( b ) − f ( k − 1 ) ( a ) ) + R ( f , m ) . {\displaystyle \sum _{k=a+1}^{b}f(k)=\sum _{k=0}^{m}{\frac {B_{k}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m).}

This form is for example the source for the important Euler–Maclaurin expansion of the zeta function

ζ ( s ) = ∑ k = 0 m B k + k ! s k − 1 ¯ + R ( s , m ) = B 0 0 ! s − 1 ¯ + B 1 + 1 ! s 0 ¯ + B 2 2 ! s 1 ¯ + ⋯ + R ( s , m ) = 1 s − 1 + 1 2 + 1 12 s + ⋯ + R ( s , m ) . {\displaystyle {\begin{aligned}\zeta (s)&=\sum _{k=0}^{m}{\frac {B_{k}^{+}}{k!}}s^{\overline {k-1}}+R(s,m)\\&={\frac {B_{0}}{0!}}s^{\overline {-1}}+{\frac {B_{1}^{+}}{1!}}s^{\overline {0}}+{\frac {B_{2}}{2!}}s^{\overline {1}}+\cdots +R(s,m)\\&={\frac {1}{s-1}}+{\frac {1}{2}}+{\frac {1}{12}}s+\cdots +R(s,m).\end{aligned}}}

Here s denotes the rising factorial power.

Bernoulli numbers are also frequently used in other kinds of asymptotic expansions. The following example is the classical Poincaré-type asymptotic expansion of the digamma function ψ.

ψ ( z ) ∼ ln ⁡ z − ∑ k = 1 ∞ B k + k z k {\displaystyle \psi (z)\sim \ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+}}{kz^{k}}}}

Sum of powers

Main article: Faulhaber's formula

Bernoulli numbers feature prominently in the closed form expression of the sum of the mth powers of the first n positive integers. For m, n ≥ 0 define

S m ( n ) = ∑ k = 1 n k m = 1 m + 2 m + ⋯ + n m . {\displaystyle S_{m}(n)=\sum _{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots +n^{m}.}

This expression can always be rewritten as a polynomial in n of degree m + 1. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:

S m ( n ) = 1 m + 1 ∑ k = 0 m ( m + 1 k ) B k + n m + 1 − k = m ! ∑ k = 0 m B k + n m + 1 − k k ! ( m + 1 − k ) ! , {\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+}n^{m+1-k}=m!\sum _{k=0}^{m}{\frac {B_{k}^{+}n^{m+1-k}}{k!(m+1-k)!}},}

where (
k) denotes the binomial coefficient.

For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, ... OEIS: A000217.

1 + 2 + ⋯ + n = 1 2 ( B 0 n 2 + 2 B 1 + n 1 ) = 1 2 ( n 2 + n ) . {\displaystyle 1+2+\cdots +n={\frac {1}{2}}(B_{0}n^{2}+2B_{1}^{+}n^{1})={\tfrac {1}{2}}(n^{2}+n).}

Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, ... OEIS: A000330.

1 2 + 2 2 + ⋯ + n 2 = 1 3 ( B 0 n 3 + 3 B 1 + n 2 + 3 B 2 n 1 ) = 1 3 ( n 3 + 3 2 n 2 + 1 2 n ) . {\displaystyle 1^{2}+2^{2}+\cdots +n^{2}={\frac {1}{3}}(B_{0}n^{3}+3B_{1}^{+}n^{2}+3B_{2}n^{1})={\tfrac {1}{3}}\left(n^{3}+{\tfrac {3}{2}}n^{2}+{\tfrac {1}{2}}n\right).}

Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:

S m ( n ) = 1 m + 1 ∑ k = 0 m ( − 1 ) k ( m + 1 k ) B k − n m + 1 − k . {\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}(-1)^{k}{\binom {m+1}{k}}B_{k}^{-{}}n^{m+1-k}.}

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sums of powers.

Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog.

Taylor series

The Bernoulli numbers appear in the Taylor series expansion of many trigonometric functions and hyperbolic functions.

tan ⁡ x = 1 x ∑ n = 1 ∞ ( − 1 ) n − 1 2 2 n ( 2 2 n − 1 ) B 2 n ( 2 n ) ! x 2 n − 1 , | x | < π 2 . cot ⁡ x = 1 x ∑ n = 0 ∞ ( − 1 ) n B 2 n ( 2 x ) 2 n ( 2 n ) ! , 0 < | x | < π . tanh ⁡ x = 1 x ∑ n = 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n ( 2 n ) ! x 2 n − 1 , | x | < π 2 . coth ⁡ x = 1 x ∑ n = 0 ∞ B 2 n ( 2 x ) 2 n ( 2 n ) ! , 0 < | x | < π . {\displaystyle {\begin{aligned}\tan x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&\left|x\right|<{\frac {\pi }{2}}.\\\cot x&={1 \over x}\sum _{n=0}^{\infty }{\frac {(-1)^{n}B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\\\tanh x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&|x|<{\frac {\pi }{2}}.\\\coth x&={1 \over x}\sum _{n=0}^{\infty }{\frac {B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\end{aligned}}}

Laurent series

The Bernoulli numbers appear in the following Laurent series:

Digamma function: ψ ( z ) = ln ⁡ z − ∑ k = 1 ∞ B k + k z k {\displaystyle \psi (z)=\ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+{}}}{kz^{k}}}}

Use in topology

The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizable manifolds involves Bernoulli numbers. Let ESn be the number of such exotic spheres for n ≥ 2, then

ES n = ( 2 2 n − 2 − 2 4 n − 3 ) Numerator ⁡ ( B 4 n 4 n ) . {\displaystyle {\textit {ES}}_{n}=(2^{2n-2}-2^{4n-3})\operatorname {Numerator} \left({\frac {B_{4n}}{4n}}\right).}

The Hirzebruch signature theorem for the L genus of a smooth oriented closed manifold of dimension 4n also involves Bernoulli numbers.

Connections with combinatorial numbers

The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the inclusion–exclusion principle.

Connection with Worpitzky numbers

The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function n! and the power function k is employed. The signless Worpitzky numbers are defined as

W n , k = ∑ v = 0 k ( − 1 ) v + k ( v + 1 ) n k ! v ! ( k − v ) ! . {\displaystyle W_{n,k}=\sum _{v=0}^{k}(-1)^{v+k}(v+1)^{n}{\frac {k!}{v!(k-v)!}}.}

They can also be expressed through the Stirling numbers of the second kind

W n , k = k ! { n + 1 k + 1 } . {\displaystyle W_{n,k}=k!\left\{{n+1 \atop k+1}\right\}.}

A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the harmonic sequence 1, ⁠1/2⁠, ⁠1/3⁠, ...

B n = ∑ k = 0 n ( − 1 ) k W n , k k + 1   =   ∑ k = 0 n 1 k + 1 ∑ v = 0 k ( − 1 ) v ( v + 1 ) n ( k v )   . {\displaystyle B_{n}=\sum _{k=0}^{n}(-1)^{k}{\frac {W_{n,k}}{k+1}}\ =\ \sum _{k=0}^{n}{\frac {1}{k+1}}\sum _{v=0}^{k}(-1)^{v}(v+1)^{n}{k \choose v}\ .}
B0 = 1
B1 = 1 − ⁠1/2⁠
B2 = 1 − ⁠3/2⁠ + ⁠2/3⁠
B3 = 1 − ⁠7/2⁠ + ⁠12/3⁠ − ⁠6/4⁠
B4 = 1 − ⁠15/2⁠ + ⁠50/3⁠ − ⁠60/4⁠ + ⁠24/5⁠
B5 = 1 − ⁠31/2⁠ + ⁠180/3⁠ − ⁠390/4⁠ + ⁠360/5⁠ − ⁠120/6⁠
B6 = 1 − ⁠63/2⁠ + ⁠602/3⁠ − ⁠2100/4⁠ + ⁠3360/5⁠ − ⁠2520/6⁠ + ⁠720/7⁠

This representation has B
1 = +⁠1/2⁠.

Consider the sequence sn, n ≥ 0. From Worpitzky's numbers OEIS: A028246, OEIS: A163626 applied to s0, s0, s1, s0, s1, s2, s0, s1, s2, s3, ... is identical to the Akiyama–Tanigawa transform applied to sn (see Connection with Stirling numbers of the first kind). This can be seen via the table:

Identity of
Worpitzky's representation and Akiyama–Tanigawa transform
1 0 1 0 0 1 0 0 0 1 0 0 0 0 1
1 −1 0 2 −2 0 0 3 −3 0 0 0 4 −4
1 −3 2 0 4 −10 6 0 0 9 −21 12
1 −7 12 −6 0 8 −38 54 −24
1 −15 50 −60 24

The first row represents s0, s1, s2, s3, s4.

Hence for the second fractional Euler numbers OEIS: A198631 (n) / OEIS: A006519 (n + 1):

E0 = 1
E1 = 1 − ⁠1/2⁠
E2 = 1 − ⁠3/2⁠ + ⁠2/4⁠
E3 = 1 − ⁠7/2⁠ + ⁠12/4⁠ − ⁠6/8⁠
E4 = 1 − ⁠15/2⁠ + ⁠50/4⁠ − ⁠60/8⁠ + ⁠24/16⁠
E5 = 1 − ⁠31/2⁠ + ⁠180/4⁠ − ⁠390/8⁠ + ⁠360/16⁠ − ⁠120/32⁠
E6 = 1 − ⁠63/2⁠ + ⁠602/4⁠ − ⁠2100/8⁠ + ⁠3360/16⁠ − ⁠2520/32⁠ + ⁠720/64⁠

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for n ≥ 1

B n = n 2 n + 1 − 2 ∑ k = 0 n − 1 ( − 2 ) − k W n − 1 , k . {\displaystyle B_{n}={\frac {n}{2^{n+1}-2}}\sum _{k=0}^{n-1}(-2)^{-k}\,W_{n-1,k}.}

The simplified second Worpitzky's representation of the second Bernoulli numbers is:

OEIS: A164555 (n + 1) / OEIS: A027642(n + 1) = ⁠n + 1/2 − 2⁠ × OEIS: A198631(n) / OEIS: A006519(n + 1)

which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:

⁠1/2⁠, ⁠1/6⁠, 0, −⁠1/30⁠, 0, ⁠1/42⁠, ... = (⁠1/2⁠, ⁠1/3⁠, ⁠3/14⁠, ⁠2/15⁠, ⁠5/62⁠, ⁠1/21⁠, ...) × (1, ⁠1/2⁠, 0, −⁠1/4⁠, 0, ⁠1/2⁠, ...)

The numerators of the first parentheses are OEIS: A111701 (see Connection with Stirling numbers of the first kind).

Connection with Stirling numbers of the second kind

If one defines the Bernoulli polynomials Bk(j) as:

B k ( j ) = k ∑ m = 0 k − 1 ( j m + 1 ) S ( k − 1 , m ) m ! + B k {\displaystyle B_{k}(j)=k\sum _{m=0}^{k-1}{\binom {j}{m+1}}S(k-1,m)m!+B_{k}}

where Bk for k = 0, 1, 2,... are the Bernoulli numbers, and S(k,m) is a Stirling number of the second kind.

One also has the following for Bernoulli polynomials,

B k ( j ) = ∑ n = 0 k ( k n ) B n j k − n . {\displaystyle B_{k}(j)=\sum _{n=0}^{k}{\binom {k}{n}}B_{n}j^{k-n}.}

The coefficient of j in (
m + 1) is ⁠(−1)/m + 1⁠.

Comparing the coefficient of j in the two expressions of Bernoulli polynomials, one has:

B k = ∑ m = 0 k − 1 ( − 1 ) m m ! m + 1 S ( k − 1 , m ) {\displaystyle B_{k}=\sum _{m=0}^{k-1}(-1)^{m}{\frac {m!}{m+1}}S(k-1,m)}

(resulting in B1 = +⁠1/2⁠) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.

Connection with Stirling numbers of the first kind

The two main formulas relating the unsigned Stirling numbers of the first kind to the Bernoulli numbers (with B1 = +⁠1/2⁠) are

1 m ! ∑ k = 0 m ( − 1 ) k [ m + 1 k + 1 ] B k = 1 m + 1 , {\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\leftB_{k}={\frac {1}{m+1}},}

and the inversion of this sum (for n ≥ 0, m ≥ 0)

1 m ! ∑ k = 0 m ( − 1 ) k [ m + 1 k + 1 ] B n + k = A n , m . {\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\leftB_{n+k}=A_{n,m}.}

Here the number An,m are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.

Akiyama–Tanigawa number
mn 0 1 2 3 4
0 1 ⁠1/2⁠ ⁠1/3⁠ ⁠1/4⁠ ⁠1/5⁠
1 ⁠1/2⁠ ⁠1/3⁠ ⁠1/4⁠ ⁠1/5⁠ ...
2 ⁠1/6⁠ ⁠1/6⁠ ⁠3/20⁠ ... ...
3 0 ⁠1/30⁠ ... ... ...
4 −⁠1/30⁠ ... ... ... ...

The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See OEIS: A051714/OEIS: A051715.

An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = OEIS: A000004, the autosequence is of the first kind. Example: OEIS: A000045, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: OEIS: A164555/OEIS: A027642, the second Bernoulli numbers (see OEIS: A190339). The Akiyama–Tanigawa transform applied to 2 = 1/OEIS: A000079 leads to OEIS: A198631 (n) / OEIS: A06519 (n + 1). Hence:

Akiyama–Tanigawa transform for the second Euler numbers
mn 0 1 2 3 4
0 1 ⁠1/2⁠ ⁠1/4⁠ ⁠1/8⁠ ⁠1/16⁠
1 ⁠1/2⁠ ⁠1/2⁠ ⁠3/8⁠ ⁠1/4⁠ ...
2 0 ⁠1/4⁠ ⁠3/8⁠ ... ...
3 −⁠1/4⁠ −⁠1/4⁠ ... ... ...
4 0 ... ... ... ...

See OEIS: A209308 and OEIS: A227577. OEIS: A198631 (n) / OEIS: A006519 (n + 1) are the second (fractional) Euler numbers and an autosequence of the second kind.

(⁠OEIS: A164555 (n + 2)/OEIS: A027642 (n + 2)⁠ = ⁠1/6⁠, 0, −⁠1/30⁠, 0, ⁠1/42⁠, ...) × ( ⁠2 − 2/n + 2⁠ = 3, ⁠14/3⁠, ⁠15/2⁠, ⁠62/5⁠, 21, ...) = ⁠OEIS: A198631 (n + 1)/OEIS: A006519 (n + 2)⁠ = ⁠1/2⁠, 0, −⁠1/4⁠, 0, ⁠1/2⁠, ....

Also valuable for OEIS: A027641 / OEIS: A027642 (see Connection with Worpitzky numbers).

Connection with Pascal's triangle

There are formulas connecting Pascal's triangle to Bernoulli numbers

B n + = | A n | ( n + 1 ) !       {\displaystyle B_{n}^{+}={\frac {|A_{n}|}{(n+1)!}}~~~}

where | A n | {\displaystyle |A_{n}|} is the determinant of a n-by-n Hessenberg matrix part of Pascal's triangle whose elements are: a i , k = { 0 if  k > 1 + i ( i + 1 k − 1 ) otherwise {\displaystyle a_{i,k}={\begin{cases}0&{\text{if }}k>1+i\\{i+1 \choose k-1}&{\text{otherwise}}\end{cases}}}

Example:

B 6 + = det ( 1 2 0 0 0 0 1 3 3 0 0 0 1 4 6 4 0 0 1 5 10 10 5 0 1 6 15 20 15 6 1 7 21 35 35 21 ) 7 ! = 120 5040 = 1 42 {\displaystyle B_{6}^{+}={\frac {\det {\begin{pmatrix}1&2&0&0&0&0\\1&3&3&0&0&0\\1&4&6&4&0&0\\1&5&10&10&5&0\\1&6&15&20&15&6\\1&7&21&35&35&21\end{pmatrix}}}{7!}}={\frac {120}{5040}}={\frac {1}{42}}}

Connection with Eulerian numbers

There are formulas connecting Eulerian numbers ⟨
m⟩ to Bernoulli numbers:

∑ m = 0 n ( − 1 ) m ⟨ n m ⟩ = 2 n + 1 ( 2 n + 1 − 1 ) B n + 1 n + 1 , ∑ m = 0 n ( − 1 ) m ⟨ n m ⟩ ( n m ) − 1 = ( n + 1 ) B n . {\displaystyle {\begin{aligned}\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle &=2^{n+1}(2^{n+1}-1){\frac {B_{n+1}}{n+1}},\\\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle {\binom {n}{m}}^{-1}&=(n+1)B_{n}.\end{aligned}}}

Both formulae are valid for n ≥ 0 if B1 is set to ⁠1/2⁠. If B1 is set to −⁠1/2⁠ they are valid only for n ≥ 1 and n ≥ 2 respectively.

A binary tree representation

The Stirling polynomials σn(x) are related to the Bernoulli numbers by Bn = n!σn(1). S. C. Woon described an algorithm to compute σn(1) as a binary tree:

Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = . Given a node N = of the tree, the left child of the node is L(N) = and the right child R(N) = . A node N = is written as ± in the initial part of the tree represented above with ± denoting the sign of a1.

Given a node N the factorial of N is defined as

N ! = a 1 ∏ k = 2 length ⁡ ( N ) a k ! . {\displaystyle N!=a_{1}\prod _{k=2}^{\operatorname {length} (N)}a_{k}!.}

Restricted to the nodes N of a fixed tree-level n the sum of ⁠1/N!⁠ is σn(1), thus

B n = ∑  tree-level  n N  node of n ! N ! . {\displaystyle B_{n}=\sum _{\stackrel {N{\text{ node of}}}{{\text{ tree-level }}n}}{\frac {n!}{N!}}.}

For example:

B1 = 1!(⁠1/2!⁠)
B2 = 2!(−⁠1/3!⁠ + ⁠1/2!2!⁠)
B3 = 3!(⁠1/4!⁠ − ⁠1/2!3!⁠ − ⁠1/3!2!⁠ + ⁠1/2!2!2!⁠)

Integral representation and continuation

The integral

b ( s ) = 2 e s i π / 2 ∫ 0 ∞ s t s 1 − e 2 π t d t t = s ! 2 s − 1 ζ ( s ) π s ( − i ) s = 2 s ! ζ ( s ) ( 2 π i ) s {\displaystyle b(s)=2e^{si\pi /2}\int _{0}^{\infty }{\frac {st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}={\frac {s!}{2^{s-1}}}{\frac {\zeta (s)}{{}\pi ^{s}{}}}(-i)^{s}={\frac {2s!\zeta (s)}{(2\pi i)^{s}}}}

has as special values b(2n) = B2n for n > 0.

For example, b(3) = ⁠3/2⁠ζ(3)πi and b(5) = −⁠15/2⁠ζ(5)πi. Here, ζ is the Riemann zeta function, and i is the imaginary unit. Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated

p = 3 2 π 3 ( 1 + 1 2 3 + 1 3 3 + ⋯ ) = 0.0581522 … q = 15 2 π 5 ( 1 + 1 2 5 + 1 3 5 + ⋯ ) = 0.0254132 … {\displaystyle {\begin{aligned}p&={\frac {3}{2\pi ^{3}}}\left(1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots \right)=0.0581522\ldots \\q&={\frac {15}{2\pi ^{5}}}\left(1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots \right)=0.0254132\ldots \end{aligned}}}

Another similar integral representation is

b ( s ) = − e s i π / 2 2 s − 1 ∫ 0 ∞ s t s sinh ⁡ π t d t t = 2 e s i π / 2 2 s − 1 ∫ 0 ∞ e π t s t s 1 − e 2 π t d t t . {\displaystyle b(s)=-{\frac {e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {st^{s}}{\sinh \pi t}}{\frac {dt}{t}}={\frac {2e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {e^{\pi t}st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}.}

The relation to the Euler numbers and π

The Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers E2n are in magnitude approximately ⁠2/π⁠(4 − 2) times larger than the Bernoulli numbers B2n. In consequence:

π ∼ 2 ( 2 2 n − 4 2 n ) B 2 n E 2 n . {\displaystyle \pi \sim 2(2^{2n}-4^{2n}){\frac {B_{2n}}{E_{2n}}}.}

This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations.

Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd n, Bn = En = 0 (with the exception B1), it suffices to consider the case when n is even.

B n = ∑ k = 0 n − 1 ( n − 1 k ) n 4 n − 2 n E k n = 2 , 4 , 6 , … E n = ∑ k = 1 n ( n k − 1 ) 2 k − 4 k k B k n = 2 , 4 , 6 , … {\displaystyle {\begin{aligned}B_{n}&=\sum _{k=0}^{n-1}{\binom {n-1}{k}}{\frac {n}{4^{n}-2^{n}}}E_{k}&n&=2,4,6,\ldots \\E_{n}&=\sum _{k=1}^{n}{\binom {n}{k-1}}{\frac {2^{k}-4^{k}}{k}}B_{k}&n&=2,4,6,\ldots \end{aligned}}}

These conversion formulas express a connection between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to π. These numbers are defined for n ≥ 1 as

S n = 2 ( 2 π ) n ∑ k = 0 ∞ ( − 1 ) k n ( 2 k + 1 ) n = 2 ( 2 π ) n lim K → ∞ ∑ k = − K K ( 4 k + 1 ) − n . {\displaystyle S_{n}=2\left({\frac {2}{\pi }}\right)^{n}\sum _{k=0}^{\infty }{\frac {(-1)^{kn}}{(2k+1)^{n}}}=2\left({\frac {2}{\pi }}\right)^{n}\lim _{K\to \infty }\sum _{k=-K}^{K}(4k+1)^{-n}.}

The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by Leonhard Euler in a landmark paper De summis serierum reciprocarum (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few of these numbers are

S n = 1 , 1 , 1 2 , 1 3 , 5 24 , 2 15 , 61 720 , 17 315 , 277 8064 , 62 2835 , … {\displaystyle S_{n}=1,1,{\frac {1}{2}},{\frac {1}{3}},{\frac {5}{24}},{\frac {2}{15}},{\frac {61}{720}},{\frac {17}{315}},{\frac {277}{8064}},{\frac {62}{2835}},\ldots } (OEIS: A099612 / OEIS: A099617)

These are the coefficients in the expansion of sec x + tan x.

The Bernoulli numbers and Euler numbers can be understood as special views of these numbers, selected from the sequence Sn and scaled for use in special applications.

B n = ( − 1 ) ⌊ n 2 ⌋ [ n  even ] n ! 2 n − 4 n S n   , n = 2 , 3 , … E n = ( − 1 ) ⌊ n 2 ⌋ [ n  even ] n ! S n + 1 n = 0 , 1 , … {\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {n!}{2^{n}-4^{n}}}\,S_{n}\ ,&n&=2,3,\ldots \\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }n!\,S_{n+1}&n&=0,1,\ldots \end{aligned}}}

The expression has the value 1 if n is even and 0 otherwise (Iverson bracket).

These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of Rn = ⁠2Sn/Sn + 1⁠ when n is even. The Rn are rational approximations to π and two successive terms always enclose the true value of π. Beginning with n = 1 the sequence starts (OEIS: A132049 / OEIS: A132050):

2 , 4 , 3 , 16 5 , 25 8 , 192 61 , 427 136 , 4352 1385 , 12465 3968 , 158720 50521 , … ⟶ π . {\displaystyle 2,4,3,{\frac {16}{5}},{\frac {25}{8}},{\frac {192}{61}},{\frac {427}{136}},{\frac {4352}{1385}},{\frac {12465}{3968}},{\frac {158720}{50521}},\ldots \quad \longrightarrow \pi .}

These rational numbers also appear in the last paragraph of Euler's paper cited above.

Consider the Akiyama–Tanigawa transform for the sequence OEIS: A046978 (n + 2) / OEIS: A016116 (n + 1):

0 1 ⁠1/2⁠ 0 −⁠1/4⁠ −⁠1/4⁠ −⁠1/8⁠ 0
1 ⁠1/2⁠ 1 ⁠3/4⁠ 0 −⁠5/8⁠ −⁠3/4⁠
2 −⁠1/2⁠ ⁠1/2⁠ ⁠9/4⁠ ⁠5/2⁠ ⁠5/8⁠
3 −1 −⁠7/2⁠ −⁠3/4⁠ ⁠15/2⁠
4 ⁠5/2⁠ −⁠11/2⁠ −⁠99/4⁠
5 8 ⁠77/2⁠
6 −⁠61/2⁠

From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −⁠1/2⁠ × OEIS: A163982.

An algorithmic view: the Seidel triangle

The sequence Sn has another unexpected yet important property: The denominators of Sn+1 divide the factorial n!. In other words: the numbers Tn = Sn + 1 n!, sometimes called Euler zigzag numbers, are integers.

T n = 1 , 1 , 1 , 2 , 5 , 16 , 61 , 272 , 1385 , 7936 , 50521 , 353792 , … n = 0 , 1 , 2 , 3 , … {\displaystyle T_{n}=1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0,1,2,3,\ldots } (OEIS: A000111). See (OEIS: A253671).

Their exponential generating function is the sum of the secant and tangent functions.

∑ n = 0 ∞ T n x n n ! = tan ⁡ ( π 4 + x 2 ) = sec ⁡ x + tan ⁡ x {\displaystyle \sum _{n=0}^{\infty }T_{n}{\frac {x^{n}}{n!}}=\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)=\sec x+\tan x} .

Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as

B n = ( − 1 ) ⌊ n 2 ⌋ [ n  even ] n 2 n − 4 n T n − 1   n ≥ 2 E n = ( − 1 ) ⌊ n 2 ⌋ [ n  even ] T n n ≥ 0 {\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {n}{2^{n}-4^{n}}}\,T_{n-1}\ &n&\geq 2\\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }T_{n}&n&\geq 0\end{aligned}}}

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers E2n are given immediately by T2n and the Bernoulli numbers B2n are fractions obtained from T2n - 1 by some easy shifting, avoiding rational arithmetic.

What remains is to find a convenient way to compute the numbers Tn. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate Tn.

1 → 1 1 2 2 1 ← → 2 4 5 5 16 16 14 10 5 ← {\displaystyle {\begin{array}{crrrcc}{}&{}&{\color {red}1}&{}&{}&{}\\{}&{\rightarrow }&{\color {blue}1}&{\color {red}1}&{}\\{}&{\color {red}2}&{\color {blue}2}&{\color {blue}1}&{\leftarrow }\\{\rightarrow }&{\color {blue}2}&{\color {blue}4}&{\color {blue}5}&{\color {red}5}\\{\color {red}16}&{\color {blue}16}&{\color {blue}14}&{\color {blue}10}&{\color {blue}5}&{\leftarrow }\end{array}}} Seidel's algorithm for Tn
  1. Start by putting 1 in row 0 and let k denote the number of the row currently being filled
  2. If k is odd, then put the number on the left end of the row k − 1 in the first position of the row k, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper
  3. At the end of the row duplicate the last number.
  4. If k is even, proceed similar in the other direction.

Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont ) and was rediscovered several times thereafter.

Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers T2n and recommended this method for computing B2n and E2n 'on electronic computers using only simple operations on integers'.

V. I. Arnold rediscovered Seidel's algorithm and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.

Triangular form:

1
1 1
2 2 1
2 4 5 5
16 16 14 10 5
16 32 46 56 61 61
272 272 256 224 178 122 61

Only OEIS: A000657, with one 1, and OEIS: A214267, with two 1s, are in the OEIS.

Distribution with a supplementary 1 and one 0 in the following rows:

1
0 1
−1 −1 0
0 −1 −2 −2
5 5 4 2 0
0 5 10 14 16 16
−61 −61 −56 −46 −32 −16 0

This is OEIS: A239005, a signed version of OEIS: A008280. The main andiagonal is OEIS: A122045. The main diagonal is OEIS: A155585. The central column is OEIS: A099023. Row sums: 1, 1, −2, −5, 16, 61.... See OEIS: A163747. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below.

The Akiyama–Tanigawa algorithm applied to OEIS: A046978 (n + 1) / OEIS: A016116(n) yields:

1 1 ⁠1/2⁠ 0 −⁠1/4⁠ −⁠1/4⁠ −⁠1/8⁠
0 1 ⁠3/2⁠ 1 0 −⁠3/4⁠
−1 −1 ⁠3/2⁠ 4 ⁠15/4⁠
0 −5 −⁠15/2⁠ 1
5 5 −⁠51/2⁠
0 61
−61

1. The first column is OEIS: A122045. Its binomial transform leads to:

1 1 0 −2 0 16 0
0 −1 −2 2 16 −16
−1 −1 4 14 −32
0 5 10 −46
5 5 −56
0 −61
−61

The first row of this array is OEIS: A155585. The absolute values of the increasing antidiagonals are OEIS: A008280. The sum of the antidiagonals is −OEIS: A163747 (n + 1).

2. The second column is 1 1 −1 −5 5 61 −61 −1385 1385.... Its binomial transform yields:

1 2 2 −4 −16 32 272
1 0 −6 −12 48 240
−1 −6 −6 60 192
−5 0 66 32
5 66 66
61 0
−61

The first row of this array is 1 2 2 −4 −16 32 272 544 −7936 15872 353792 −707584.... The absolute values of the second bisection are the double of the absolute values of the first bisection.

Consider the Akiyama-Tanigawa algorithm applied to OEIS: A046978 (n) / (OEIS: A158780 (n + 1) = abs(OEIS: A117575 (n)) + 1 = 1, 2, 2, ⁠3/2⁠, 1, ⁠3/4⁠, ⁠3/4⁠, ⁠7/8⁠, 1, ⁠17/16⁠, ⁠17/16⁠, ⁠33/32⁠....

1 2 2 ⁠3/2⁠ 1 ⁠3/4⁠ ⁠3/4⁠
−1 0 ⁠3/2⁠ 2 ⁠5/4⁠ 0
−1 −3 −⁠3/2⁠ 3 ⁠25/4⁠
2 −3 −⁠27/2⁠ −13
5 21 −⁠3/2⁠
−16 45
−61

The first column whose the absolute values are OEIS: A000111 could be the numerator of a trigonometric function.

OEIS: A163747 is an autosequence of the first kind (the main diagonal is OEIS: A000004). The corresponding array is:

0 −1 −1 2 5 −16 −61
−1 0 3 3 −21 −45
1 3 0 −24 −24
2 −3 −24 0
−5 −21 24
−16 45
−61

The first two upper diagonals are −1 3 −24 402... = (−1) × OEIS: A002832. The sum of the antidiagonals is 0 −2 0 10... = 2 × OEIS: A122045(n + 1).

−OEIS: A163982 is an autosequence of the second kind, like for instance OEIS: A164555 / OEIS: A027642. Hence the array:

2 1 −1 −2 5 16 −61
−1 −2 −1 7 11 −77
−1 1 8 4 −88
2 7 −4 −92
5 −11 −88
−16 −77
−61

The main diagonal, here 2 −2 8 −92..., is the double of the first upper one, here OEIS: A099023. The sum of the antidiagonals is 2 0 −4 0... = 2 × OEIS: A155585(n + 1). OEIS: A163747 − OEIS: A163982 = 2 × OEIS: A122045.

A combinatorial view: alternating permutations

Main article: Alternating permutations

Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis. Looking at the first terms of the Taylor expansion of the trigonometric functions tan x and sec x André made a startling discovery.

tan ⁡ x = x + 2 x 3 3 ! + 16 x 5 5 ! + 272 x 7 7 ! + 7936 x 9 9 ! + ⋯ sec ⁡ x = 1 + x 2 2 ! + 5 x 4 4 ! + 61 x 6 6 ! + 1385 x 8 8 ! + 50521 x 10 10 ! + ⋯ {\displaystyle {\begin{aligned}\tan x&=x+{\frac {2x^{3}}{3!}}+{\frac {16x^{5}}{5!}}+{\frac {272x^{7}}{7!}}+{\frac {7936x^{9}}{9!}}+\cdots \\\sec x&=1+{\frac {x^{2}}{2!}}+{\frac {5x^{4}}{4!}}+{\frac {61x^{6}}{6!}}+{\frac {1385x^{8}}{8!}}+{\frac {50521x^{10}}{10!}}+\cdots \end{aligned}}}

The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of tan x + sec x has as coefficients the rational numbers Sn.

tan ⁡ x + sec ⁡ x = 1 + x + 1 2 x 2 + 1 3 x 3 + 5 24 x 4 + 2 15 x 5 + 61 720 x 6 + ⋯ {\displaystyle \tan x+\sec x=1+x+{\tfrac {1}{2}}x^{2}+{\tfrac {1}{3}}x^{3}+{\tfrac {5}{24}}x^{4}+{\tfrac {2}{15}}x^{5}+{\tfrac {61}{720}}x^{6}+\cdots }

André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).

Related sequences

The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: B0 = 1, B1 = 0, B2 = ⁠1/6⁠, B3 = 0, B4 = −⁠1/30⁠, OEIS: A176327 / OEIS: A027642. Via the second row of its inverse Akiyama–Tanigawa transform OEIS: A177427, they lead to Balmer series OEIS: A061037 / OEIS: A061038.

The Akiyama–Tanigawa algorithm applied to OEIS: A060819 (n + 4) / OEIS: A145979 (n) leads to the Bernoulli numbers OEIS: A027641 / OEIS: A027642, OEIS: A164555 / OEIS: A027642, or OEIS: A176327 OEIS: A176289 without B1, named intrinsic Bernoulli numbers Bi(n).

1 ⁠5/6⁠ ⁠3/4⁠ ⁠7/10⁠ ⁠2/3⁠
⁠1/6⁠ ⁠1/6⁠ ⁠3/20⁠ ⁠2/15⁠ ⁠5/42⁠
0 ⁠1/30⁠ ⁠1/20⁠ ⁠2/35⁠ ⁠5/84⁠
−⁠1/30⁠ −⁠1/30⁠ −⁠3/140⁠ −⁠1/105⁠ 0
0 −⁠1/42⁠ −⁠1/28⁠ −⁠4/105⁠ −⁠1/28⁠

Hence another link between the intrinsic Bernoulli numbers and the Balmer series via OEIS: A145979 (n).

OEIS: A145979 (n − 2) = 0, 2, 1, 6,... is a permutation of the non-negative numbers.

The terms of the first row are f(n) = ⁠1/2⁠ + ⁠1/n + 2⁠. 2, f(n) is an autosequence of the second kind. 3/2, f(n) leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5 ... = 1/2 + log 2.

Consider g(n) = 1/2 – 1 / (n+2) = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives:

0 ⁠1/6⁠ ⁠1/4⁠ ⁠3/10⁠ ⁠1/3⁠ ⁠5/14⁠ ...
−⁠1/6⁠ −⁠1/6⁠ −⁠3/20⁠ −⁠2/15⁠ −⁠5/42⁠ −⁠3/28⁠ ...
0 −⁠1/30⁠ −⁠1/20⁠ −⁠2/35⁠ −⁠5/84⁠ −⁠5/84⁠ ...
⁠1/30⁠ ⁠1/30⁠ ⁠3/140⁠ ⁠1/105⁠ 0 −⁠1/140⁠ ...

0, g(n), is an autosequence of the second kind.

Euler OEIS: A198631 (n) / OEIS: A006519 (n + 1) without the second term (⁠1/2⁠) are the fractional intrinsic Euler numbers Ei(n) = 1, 0, −⁠1/4⁠, 0, ⁠1/2⁠, 0, −⁠17/8⁠, 0, ... The corresponding Akiyama transform is:

1 1 ⁠7/8⁠ ⁠3/4⁠ ⁠21/32⁠
0 ⁠1/4⁠ ⁠3/8⁠ ⁠3/8⁠ ⁠5/16⁠
−⁠1/4⁠ −⁠1/4⁠ 0 ⁠1/4⁠ ⁠25/64⁠
0 −⁠1/2⁠ −⁠3/4⁠ −⁠9/16⁠ −⁠5/32⁠
⁠1/2⁠ ⁠1/2⁠ −⁠9/16⁠ −⁠13/8⁠ −⁠125/64⁠

The first line is Eu(n). Eu(n) preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are OEIS: A069834 preceded by 0. The difference table is:

0 1 1 ⁠7/8⁠ ⁠3/4⁠ ⁠21/32⁠ ⁠19/32⁠
1 0 −⁠1/8⁠ −⁠1/8⁠ −⁠3/32⁠ −⁠1/16⁠ −⁠5/128⁠
−1 −⁠1/8⁠ 0 ⁠1/32⁠ ⁠1/32⁠ ⁠3/128⁠ ⁠1/64⁠

Arithmetical properties of the Bernoulli numbers

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = −nζ(1 − n) for integers n ≥ 0 provided for n = 0 the expression −nζ(1 − n) is understood as the limiting value and the convention B1 = ⁠1/2⁠ is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that p is a prime number if and only if pBp − 1 is congruent to −1 modulo p. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

The Kummer theorems

The Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem, which says:

If the odd prime p does not divide any of the numerators of the Bernoulli numbers B2, B4, ..., Bp − 3 then x + y + z = 0 has no solutions in nonzero integers.

Prime numbers with this property are called regular primes. Another classical result of Kummer are the following congruences.

Main article: Kummer's congruence
Let p be an odd prime and b an even number such that p − 1 does not divide b. Then for any non-negative integer k
B k ( p − 1 ) + b k ( p − 1 ) + b ≡ B b b ( mod p ) . {\displaystyle {\frac {B_{k(p-1)+b}}{k(p-1)+b}}\equiv {\frac {B_{b}}{b}}{\pmod {p}}.}

A generalization of these congruences goes by the name of p-adic continuity.

p-adic continuity

If b, m and n are positive integers such that m and n are not divisible by p − 1 and m ≡ n (mod p (p − 1)), then

( 1 − p m − 1 ) B m m ≡ ( 1 − p n − 1 ) B n n ( mod p b ) . {\displaystyle (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}

Since Bn = −nζ(1 − n), this can also be written

( 1 − p − u ) ζ ( u ) ≡ ( 1 − p − v ) ζ ( v ) ( mod p b ) , {\displaystyle \left(1-p^{-u}\right)\zeta (u)\equiv \left(1-p^{-v}\right)\zeta (v){\pmod {p^{b}}},}

where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 modulo p − 1. This tells us that the Riemann zeta function, with 1 − p taken out of the Euler product formula, is continuous in the p-adic numbers on odd negative integers congruent modulo p − 1 to a particular a ≢ 1 mod (p − 1), and so can be extended to a continuous function ζp(s) for all p-adic integers Z p , {\displaystyle \mathbb {Z} _{p},} the p-adic zeta function.

Ramanujan's congruences

The following relations, due to Ramanujan, provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition:

( m + 3 m ) B m = { m + 3 3 − ∑ j = 1 m 6 ( m + 3 m − 6 j ) B m − 6 j , if  m ≡ 0 ( mod 6 ) ; m + 3 3 − ∑ j = 1 m − 2 6 ( m + 3 m − 6 j ) B m − 6 j , if  m ≡ 2 ( mod 6 ) ; − m + 3 6 − ∑ j = 1 m − 4 6 ( m + 3 m − 6 j ) B m − 6 j , if  m ≡ 4 ( mod 6 ) . {\displaystyle {\binom {m+3}{m}}B_{m}={\begin{cases}{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 0{\pmod {6}};\\{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m-2}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 2{\pmod {6}};\\-{\frac {m+3}{6}}-\sum \limits _{j=1}^{\frac {m-4}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 4{\pmod {6}}.\end{cases}}}

Von Staudt–Clausen theorem

Main article: Von Staudt–Clausen theorem

The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt and Thomas Clausen independently in 1840. The theorem states that for every n > 0,

B 2 n + ∑ ( p − 1 ) ∣ 2 n 1 p {\displaystyle B_{2n}+\sum _{(p-1)\,\mid \,2n}{\frac {1}{p}}}

is an integer. The sum extends over all primes p for which p − 1 divides 2n.

A consequence of this is that the denominator of B2n is given by the product of all primes p for which p − 1 divides 2n. In particular, these denominators are square-free and divisible by 6.

Why do the odd Bernoulli numbers vanish?

The sum

φ k ( n ) = ∑ i = 0 n i k − n k 2 {\displaystyle \varphi _{k}(n)=\sum _{i=0}^{n}i^{k}-{\frac {n^{k}}{2}}}

can be evaluated for negative values of the index n. Doing so will show that it is an odd function for even values of k, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that B2k + 1 − m is 0 for m even and 2k + 1 − m > 1; and that the term for B1 is cancelled by the subtraction. The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for n > 1).

From the von Staudt–Clausen theorem it is known that for odd n > 1 the number 2Bn is an integer. This seems trivial if one knows beforehand that the integer in question is zero. However, by applying Worpitzky's representation one gets

2 B n = ∑ m = 0 n ( − 1 ) m 2 m + 1 m ! { n + 1 m + 1 } = 0 ( n > 1  is odd ) {\displaystyle 2B_{n}=\sum _{m=0}^{n}(-1)^{m}{\frac {2}{m+1}}m!\left\{{n+1 \atop m+1}\right\}=0\quad (n>1{\text{ is odd}})}

as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let Sn,m be the number of surjective maps from {1, 2, ..., n} to {1, 2, ..., m}, then Sn,m = m!{
m}. The last equation can only hold if

∑ odd  m = 1 n − 1 2 m 2 S n , m = ∑ even  m = 2 n 2 m 2 S n , m ( n > 2  is even ) . {\displaystyle \sum _{{\text{odd }}m=1}^{n-1}{\frac {2}{m^{2}}}S_{n,m}=\sum _{{\text{even }}m=2}^{n}{\frac {2}{m^{2}}}S_{n,m}\quad (n>2{\text{ is even}}).}

This equation can be proved by induction. The first two examples of this equation are

n = 4: 2 + 8 = 7 + 3,
n = 6: 2 + 120 + 144 = 31 + 195 + 40.

Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.

A restatement of the Riemann hypothesis

The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the Riemann hypothesis (RH) which uses only the Bernoulli numbers. In fact Marcel Riesz proved that the RH is equivalent to the following assertion:

For every ε > ⁠1/4⁠ there exists a constant Cε > 0 (depending on ε) such that |R(x)| < Cεx as x → ∞.

Here R(x) is the Riesz function

R ( x ) = 2 ∑ k = 1 ∞ k k ¯ x k ( 2 π ) 2 k ( B 2 k 2 k ) = 2 ∑ k = 1 ∞ k k ¯ x k ( 2 π ) 2 k β 2 k . {\displaystyle R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left({\frac {B_{2k}}{2k}}\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.}

n denotes the rising factorial power in the notation of D. E. Knuth. The numbers βn = ⁠Bn/n⁠ occur frequently in the study of the zeta function and are significant because βn is a p-integer for primes p where p − 1 does not divide n. The βn are called divided Bernoulli numbers.

Generalized Bernoulli numbers

The generalized Bernoulli numbers are certain algebraic numbers, defined similarly to the Bernoulli numbers, that are related to special values of Dirichlet L-functions in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.

Let χ be a Dirichlet character modulo f. The generalized Bernoulli numbers attached to χ are defined by

∑ a = 1 f χ ( a ) t e a t e f t − 1 = ∑ k = 0 ∞ B k , χ t k k ! . {\displaystyle \sum _{a=1}^{f}\chi (a){\frac {te^{at}}{e^{ft}-1}}=\sum _{k=0}^{\infty }B_{k,\chi }{\frac {t^{k}}{k!}}.}

Apart from the exceptional B1,1 = ⁠1/2⁠, we have, for any Dirichlet character χ, that Bk,χ = 0 if χ(−1) ≠ (−1).

Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers k ≥ 1:

L ( 1 − k , χ ) = − B k , χ k , {\displaystyle L(1-k,\chi )=-{\frac {B_{k,\chi }}{k}},}

where L(s,χ) is the Dirichlet L-function of χ.

Eisenstein–Kronecker number

Main article: Eisenstein–Kronecker number

Eisenstein–Kronecker numbers are an analogue of the generalized Bernoulli numbers for imaginary quadratic fields. They are related to critical L-values of Hecke characters.

Appendix

Assorted identities

  • Umbral calculus gives a compact form of Bernoulli's formula by using an abstract symbol B:
    S m ( n ) = 1 m + 1 ( ( B + n ) m + 1 − B m + 1 ) {\displaystyle S_{m}(n)={\frac {1}{m+1}}((\mathbf {B} +n)^{m+1}-B_{m+1})}

    where the symbol B that appears during binomial expansion of the parenthesized term is to be replaced by the Bernoulli number Bk (and B1 = +⁠1/2⁠). More suggestively and mnemonically, this may be written as a definite integral:

    S m ( n ) = ∫ 0 n ( B + x ) m d x {\displaystyle S_{m}(n)=\int _{0}^{n}(\mathbf {B} +x)^{m}\,dx}

    Many other Bernoulli identities can be written compactly with this symbol, e.g.

    ( 1 − 2 B ) m = ( 2 − 2 m ) B m {\displaystyle (1-2\mathbf {B} )^{m}=(2-2^{m})B_{m}}
  • Let n be non-negative and even
    ζ ( n ) = ( − 1 ) n 2 − 1 B n ( 2 π ) n 2 ( n ! ) {\displaystyle \zeta (n)={\frac {(-1)^{{\frac {n}{2}}-1}B_{n}(2\pi )^{n}}{2(n!)}}}
  • The nth cumulant of the uniform probability distribution on the interval is ⁠Bn/n⁠.
  • Let n? = ⁠1/n!⁠ and n ≥ 1. Then Bn is the following (n + 1) × (n + 1) determinant:
    B n = n ! | 1 0 ⋯ 0 1 2 ? 1 ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ n ? ( n − 1 ) ? ⋯ 1 0 ( n + 1 ) ? n ? ⋯ 2 ? 0 | = n ! | 1 0 ⋯ 0 1 1 2 ! 1 ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ 1 n ! 1 ( n − 1 ) ! ⋯ 1 0 1 ( n + 1 ) ! 1 n ! ⋯ 1 2 ! 0 | {\displaystyle {\begin{aligned}B_{n}&=n!{\begin{vmatrix}1&0&\cdots &0&1\\2?&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\n?&(n-1)?&\cdots &1&0\\(n+1)?&n?&\cdots &2?&0\end{vmatrix}}\\&=n!{\begin{vmatrix}1&0&\cdots &0&1\\{\frac {1}{2!}}&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{n!}}&{\frac {1}{(n-1)!}}&\cdots &1&0\\{\frac {1}{(n+1)!}}&{\frac {1}{n!}}&\cdots &{\frac {1}{2!}}&0\end{vmatrix}}\end{aligned}}}
    Thus the determinant is σn(1), the Stirling polynomial at x = 1.
  • For even-numbered Bernoulli numbers, B2p is given by the (p + 1) × (p + 1) determinant::
    B 2 p = − ( 2 p ) ! 2 2 p − 2 | 1 0 0 ⋯ 0 1 1 3 ! 1 0 ⋯ 0 0 1 5 ! 1 3 ! 1 ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 1 ( 2 p + 1 ) ! 1 ( 2 p − 1 ) ! 1 ( 2 p − 3 ) ! ⋯ 1 3 ! 0 | {\displaystyle B_{2p}=-{\frac {(2p)!}{2^{2p}-2}}{\begin{vmatrix}1&0&0&\cdots &0&1\\{\frac {1}{3!}}&1&0&\cdots &0&0\\{\frac {1}{5!}}&{\frac {1}{3!}}&1&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{(2p+1)!}}&{\frac {1}{(2p-1)!}}&{\frac {1}{(2p-3)!}}&\cdots &{\frac {1}{3!}}&0\end{vmatrix}}}
  • Let n ≥ 1. Then (Leonhard Euler)
    1 n ∑ k = 1 n ( n k ) B k B n − k + B n − 1 = − B n {\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\binom {n}{k}}B_{k}B_{n-k}+B_{n-1}=-B_{n}}
  • Let n ≥ 1. Then
    ∑ k = 0 n ( n + 1 k ) ( n + k + 1 ) B n + k = 0 {\displaystyle \sum _{k=0}^{n}{\binom {n+1}{k}}(n+k+1)B_{n+k}=0}
  • Let n ≥ 0. Then (Leopold Kronecker 1883)
    B n = − ∑ k = 1 n + 1 ( − 1 ) k k ( n + 1 k ) ∑ j = 1 k j n {\displaystyle B_{n}=-\sum _{k=1}^{n+1}{\frac {(-1)^{k}}{k}}{\binom {n+1}{k}}\sum _{j=1}^{k}j^{n}}
  • Let n ≥ 1 and m ≥ 1. Then
    ( − 1 ) m ∑ r = 0 m ( m r ) B n + r = ( − 1 ) n ∑ s = 0 n ( n s ) B m + s {\displaystyle (-1)^{m}\sum _{r=0}^{m}{\binom {m}{r}}B_{n+r}=(-1)^{n}\sum _{s=0}^{n}{\binom {n}{s}}B_{m+s}}
  • Let n ≥ 4 and
    H n = ∑ k = 1 n k − 1 {\displaystyle H_{n}=\sum _{k=1}^{n}k^{-1}}
    the harmonic number. Then (H. Miki 1978)
    n 2 ∑ k = 2 n − 2 B n − k n − k B k k − ∑ k = 2 n − 2 ( n k ) B n − k n − k B k = H n B n {\displaystyle {\frac {n}{2}}\sum _{k=2}^{n-2}{\frac {B_{n-k}}{n-k}}{\frac {B_{k}}{k}}-\sum _{k=2}^{n-2}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}=H_{n}B_{n}}
  • Let n ≥ 4. Yuri Matiyasevich found (1997)
    ( n + 2 ) ∑ k = 2 n − 2 B k B n − k − 2 ∑ l = 2 n − 2 ( n + 2 l ) B l B n − l = n ( n + 1 ) B n {\displaystyle (n+2)\sum _{k=2}^{n-2}B_{k}B_{n-k}-2\sum _{l=2}^{n-2}{\binom {n+2}{l}}B_{l}B_{n-l}=n(n+1)B_{n}}
  • Faber–Pandharipande–Zagier–Gessel identity: for n ≥ 1,
    n 2 ( B n − 1 ( x ) + ∑ k = 1 n − 1 B k ( x ) k B n − k ( x ) n − k ) − ∑ k = 0 n − 1 ( n k ) B n − k n − k B k ( x ) = H n − 1 B n ( x ) . {\displaystyle {\frac {n}{2}}\left(B_{n-1}(x)+\sum _{k=1}^{n-1}{\frac {B_{k}(x)}{k}}{\frac {B_{n-k}(x)}{n-k}}\right)-\sum _{k=0}^{n-1}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}(x)=H_{n-1}B_{n}(x).}
    Choosing x = 0 or x = 1 results in the Bernoulli number identity in one or another convention.
  • The next formula is true for n ≥ 0 if B1 = B1(1) = ⁠1/2⁠, but only for n ≥ 1 if B1 = B1(0) = −⁠1/2⁠.
    ∑ k = 0 n ( n k ) B k n − k + 2 = B n + 1 n + 1 {\displaystyle \sum _{k=0}^{n}{\binom {n}{k}}{\frac {B_{k}}{n-k+2}}={\frac {B_{n+1}}{n+1}}}
  • Let n ≥ 0. Then
    − 1 + ∑ k = 0 n ( n k ) 2 n − k + 1 n − k + 1 B k ( 1 ) = 2 n {\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(1)=2^{n}}
    and
    − 1 + ∑ k = 0 n ( n k ) 2 n − k + 1 n − k + 1 B k ( 0 ) = δ n , 0 {\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(0)=\delta _{n,0}}
  • A reciprocity relation of M. B. Gelfand:
    ( − 1 ) m + 1 ∑ j = 0 k ( k j ) B m + 1 + j m + 1 + j + ( − 1 ) k + 1 ∑ j = 0 m ( m j ) B k + 1 + j k + 1 + j = k ! m ! ( k + m + 1 ) ! {\displaystyle (-1)^{m+1}\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{m+1+j}}{m+1+j}}+(-1)^{k+1}\sum _{j=0}^{m}{\binom {m}{j}}{\frac {B_{k+1+j}}{k+1+j}}={\frac {k!m!}{(k+m+1)!}}}

See also

Notes

  1. Translation of the text: " ... And if to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list:
    Sums of powers
    ∫ n = ∑ k = 1 n k = 1 2 n 2 + 1 2 n {\displaystyle \textstyle \int n=\sum _{k=1}^{n}k={\frac {1}{2}}n^{2}+{\frac {1}{2}}n}
    ⋮
    ∫ n 10 = ∑ k = 1 n k 10 = 1 11 n 11 + 1 2 n 10 + 5 6 n 9 − 1 n 7 + 1 n 5 − 1 2 n 3 + 5 66 n {\displaystyle \textstyle \int n^{10}=\sum _{k=1}^{n}k^{10}={\frac {1}{11}}n^{11}+{\frac {1}{2}}n^{10}+{\frac {5}{6}}n^{9}-1n^{7}+1n^{5}-{\frac {1}{2}}n^{3}+{\frac {5}{66}}n}
    Indeed one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these circuitous computations: For c {\displaystyle \textstyle c} is taken as the exponent of any power, the sum of all n c {\displaystyle \textstyle n^{c}} is produced or
    ∫ n c = ∑ k = 1 n k c = 1 c + 1 n c + 1 + 1 2 n c + c 2 A n c − 1 + c ( c − 1 ) ( c − 2 ) 2 ⋅ 3 ⋅ 4 B n c − 3 + c ( c − 1 ) ( c − 2 ) ( c − 3 ) ( c − 4 ) 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 C n c − 5 + c ( c − 1 ) ( c − 2 ) ( c − 3 ) ( c − 4 ) ( c − 5 ) ( c − 6 ) 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 ⋅ 8 D n c − 7 + ⋯ {\displaystyle \textstyle \int n^{c}=\sum _{k=1}^{n}k^{c}={\frac {1}{c+1}}n^{c+1}+{\frac {1}{2}}n^{c}+{\frac {c}{2}}An^{c-1}+{\frac {c(c-1)(c-2)}{2\cdot 3\cdot 4}}Bn^{c-3}+{\frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4\cdot 5\cdot 6}}Cn^{c-5}+{\frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}}Dn^{c-7}+\cdots }
    and so forth, the exponent of its power n {\displaystyle n} continually diminishing by 2 until it arrives at n {\displaystyle n} or n 2 {\displaystyle n^{2}} . The capital letters A , B , C , D , {\displaystyle \textstyle A,B,C,D,} etc. denote in order the coefficients of the last terms for ∫ n 2 , ∫ n 4 , ∫ n 6 , ∫ n 8 {\displaystyle \textstyle \int n^{2},\int n^{4},\int n^{6},\int n^{8}} , etc. namely
    A = 1 6 , B = − 1 30 , C = 1 42 , D = − 1 30 {\displaystyle \textstyle A={\frac {1}{6}},B=-{\frac {1}{30}},C={\frac {1}{42}},D=-{\frac {1}{30}}} ."
  2. The Mathematics Genealogy Project (n.d.) shows Leibniz as the academic advisor of Jakob Bernoulli. See also Miller (2017).
  3. this formula was discovered (or perhaps rediscovered) by Giorgio Pietrocola. His demonstration is available in Italian language (Pietrocola 2008).

References

  1. ^ Weisstein, Eric W., "Bernoulli Number", MathWorld
  2. ^ Selin, Helaine, ed. (1997), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, p. 819 (p. 891), Bibcode:2008ehst.book.....S, ISBN 0-7923-4066-3
  3. Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, Open Court publishing company, p. 108, ISBN 9780486434827
  4. Kitagawa, Tomoko L. (2021-07-23), "The Origin of the Bernoulli Numbers: Mathematics in Basel and Edo in the Early Eighteenth Century", The Mathematical Intelligencer, 44: 46–56, doi:10.1007/s00283-021-10072-y, ISSN 0343-6993
  5. Menabrea, L.F. (1842), "Sketch of the Analytic Engine invented by Charles Babbage, with notes upon the Memoir by the Translator Ada Augusta, Countess of Lovelace", Bibliothèque Universelle de Genève, 82, See Note G
  6. Arfken (1970), p. 278.
  7. Donald Knuth (2022), Recent News (2022): Concrete Mathematics and Bernoulli.

    But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half. He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used. It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.

  8. Peter Luschny (2013), The Bernoulli Manifesto
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