Bernoulli number

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 NIST 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 Donald KnuthDonald Knuth (2022), [https://www-cs-faculty.stanford.edu/~knuth/news22.html Recent News (2022): Concrete Mathematics and Bernoulli]. {{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.

}} following Peter Luschny's "Bernoulli Manifesto".Peter Luschny (2013), [http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html The Bernoulli Manifesto]

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

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|2n}} }}. This article does not follow that notation.

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.

File:Seki Kowa Katsuyo Sampo Bernoulli numbers.png

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 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 {{math|(n+1)^k+1}} to the sums of the {{math|p}}th powers of the first {{math|n}} positive integers for {{math|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 {{math|B₀, B₁, B₂,...}} which provide a uniform formula for all sums of powers.{{sfnp|Knuth|1993}}

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:

:"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.{{r|Selin1997_891}} 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{{sfnp|Knuth|1993}} a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834.{{r|Jacobi1834}} 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 {{math|B₀, B₁, B₂,}} ... would provide a uniform

::$\backslash sum\; n^m\; =\; \backslash frac\; 1\{m+1\}\backslash left(\; B\_0n^\{m+1\}-\backslash binom\{m+1\}\; 1\; B\_1\; n^m+\backslash binom\{m+1\}\; 2B\_2n^\{m-1\}-\backslash cdots\; +(-1)^m\backslash binom\{m+1\}mB\_mn\backslash right)$

: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 {{math|Σ n^m}} from polynomials in {{mvar|N}} to polynomials in {{mvar|n}}."{{sfnp|Knuth|1993|p=14}}

In the above Knuth meant $B\_1^-$; instead using $B\_1^+$ the formula avoids subtraction:

:$\backslash sum\; n^m\; =\; \backslash frac\; 1\{m+1\}\backslash left(\; B\_0n^\{m+1\}+\backslash binom\{m+1\}\; 1\; B^+\_1\; n^m+\backslash binom\{m+1\}\; 2B\_2n^\{m-1\}+\backslash cdots+\backslash binom\{m+1\}mB\_mn\backslash right).$

[[File:JakobBernoulliSummaePotestatum.png|thumb|right|upright=1.5|Jakob Bernoulli's "Summae Potestatum", 1713{{efn|Translation of the text:

" ... And if [one were] to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list:

Sums of powers

$\backslash textstyle\; \backslash int\; n\; =\; \backslash sum\_\{k=1\}^n\; k\; =\; \backslash frac\; \{1\}\{2\}\; n^2\; +\; \backslash frac\; \{1\}\{2\}\; n$

::::⋮

$\backslash textstyle\; \backslash int\; n^\{10\}\; =\; \backslash sum\_\{k=1\}^n\; k^\{10\}\; =\; \backslash frac\; \{1\}\{11\}\; n^\{11\}\; +\; \backslash frac\; \{1\}\{2\}\; n^\{10\}\; +\; \backslash frac\; \{5\}\{6\}\; n^9\; -\; 1\; n^7\; +\; 1\; n^5\; -\; \backslash frac\; \{1\}\{2\}\; n^3\; +\; \backslash frac\; \{5\}\{66\}\; n$

Indeed [if] 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 [if] $\backslash textstyle\; c$ is taken as the exponent of any power, the sum of all $\backslash textstyle\; n^c$ is produced or

$\backslash textstyle\; \backslash int\; n^c\; =\; \backslash sum\_\{k=1\}^n\; k^c\; =\; \backslash frac\; \{1\}\{c+1\}\; n^\{c+1\}\; +\; \backslash frac\; \{1\}\{2\}\; n^c\; +\; \backslash frac\; \{c\}\{2\}\; An^\{c-1\}\; +\; \backslash frac\; \{c(c-1)(c-2)\}\{2\backslash cdot\; 3\backslash cdot4\}\; Bn^\{c-3\}\; +\; \backslash frac\; \{c(c-1)(c-2)(c-3)(c-4)\}\{2\backslash cdot\; 3\backslash cdot\; 4\; \backslash cdot\; 5\; \backslash cdot\; 6\}\; Cn^\{c-5\}\; +\; \backslash frac\; \{c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)\}\{2\backslash cdot\; 3\backslash cdot\; 4\; \backslash cdot\; 5\; \backslash cdot\; 6\; \backslash cdot\; 7\; \backslash cdot\; 8\}\; Dn^\{c-7\}\; +\; \backslash cdots$

and so forth, the exponent of its power $n$ continually diminishing by 2 until it arrives at $n$ or $n^2$. The capital letters $\backslash textstyle\; A,\; B,\; C,\; D,$ etc. denote in order the coefficients of the last terms for $\backslash textstyle\; \backslash int\; n^2\; ,\; \backslash int\; n^4\; ,\; \backslash int\; n^6\; ,\; \backslash int\; n^8$, etc. namely

$\backslash textstyle\; A\; =\; \backslash frac\; \{1\}\{6\}\; ,\; B\; =\; -\; \backslash frac\; \{1\}\{30\}\; ,\; C\; =\; \backslash frac\; \{1\}\{42\}\; ,\; D\; =\; -\; \backslash frac\; \{1\}\{30\}$."

[Note: The text of the illustration contains some typos: ensperexit should read inspexerit, ambabimus should read ambagibus, quosque should read quousque, and in Bernoulli's original text Sumtâ should read Sumptâ or Sumptam.]

• {{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 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 {{math|A}}, {{math|B}}, {{math|C}} and {{math|D}} by Bernoulli are mapped to the notation which is now prevalent as {{math|A {{=}} B₂}}, {{math|B {{=}} B₄}}, {{math|C {{=}} B₆}}, {{math|D {{=}} B₈}}. 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 falling factorial powers {{math|c^{{{underline|k}}}}}. 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 Gottfried Wilhelm Leibniz 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 advisor of Jakob Bernoulli. See also {{harvp|Miller|2017}}.}} The letter {{math|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 {{math|1, 2, ..., n}}. Putting things together, for positive {{math|c}}, today a mathematician is likely to write Bernoulli's formula as:

: $\backslash sum\_\{k=1\}^n\; k^c\; =\; \backslash frac\{n^\{c+1\}\}\{c+1\}+\backslash frac\; 1\; 2\; n^c+\backslash sum\_\{k=2\}^c\; \backslash frac\{B\_k\}\{k!\}\; c^\{\backslash underline\{k-1\}\}n^\{c-k+1\}.$

This formula suggests setting {{math|B₁ {{=}} {{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 falling factorial {{math|c^{{{underline|k−1}}}}} 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

: $\backslash sum\_\{k=1\}^n\; k^c\; =\; \backslash sum\_\{k=0\}^c\; \backslash frac\{B\_k\}\{k!\}c^\{\backslash underline\{k-1\}\}\; n^\{c-k+1\}$

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

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

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, see {{harvp|Ireland|Rosen|1990}} or {{harvp|Conway|Guy|1996}}.

The Bernoulli numbers obey the sum formulas{{r|Weisstein2016}}

:

where $m=0,1,2...$ and {{math|δ}} denotes the Kronecker delta.

The first of these is sometimes writtenJordan (1950) p 233 as the formula (for m > 1)

$$(B+1)^m-B\_m=0,$$

where the power is expanded formally using the binomial theorem and $B^k$ is replaced by $B\_k$.

Solving for $B^\{\backslash mp\{\}\}\_m$ gives the recursive formulasIreland and Rosen (1990) p 229

: $\backslash begin\{align\}$

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{align}

In 1893 Louis Saalschütz 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 $m\backslash geq\; 1$):

:$\backslash begin\{align\}$

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}

The exponential generating functions are

:$\backslash begin\{alignat\}\{3\}$

\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}

where the substitution is $t\; \backslash to\; -\; t$. The two generating functions only differ by t.

{{collapse top|title=Proof}}

If we let $F(t)=\backslash sum\_\{i=1\}^\backslash infty\; f\_it^i$ and $G(t)=1/(1+F(t))=\backslash sum\_\{i=0\}^\backslash infty\; g\_it^i$ then

:$G(t)=1-F(t)G(t).$

Then $g\_0=1$ and for $m>0$ the m{{sup|th}} term in the series for $G(t)$ is:

:$g\_mt^i=-\backslash sum\_\{j=0\}^\{m-1\}f\_\{m-j\}g\_jt^m$

If

:$F(t)=\backslash frac\{e^t-1\}t-1=\backslash sum\_\{i=1\}^\backslash infty\; \backslash frac\{t^i\}\{(i+1)!\}$

then we find that

:$G(t)=t/(e^t-1)$

:$\backslash begin\{align\}$

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}jj!g_j\\

\end{align}

showing that the values of $i!g\_i$ obey the recursive formula for the Bernoulli numbers $B^-\_i$.

{{collapse bottom}}

The (ordinary) generating function

: $z^\{-1\}\; \backslash psi\_1(z^\{-1\})\; =\; \backslash sum\_\{m=0\}^\{\backslash infty\}\; B^+\_m\; z^m$

is an asymptotic series. It contains the trigamma function {{math|ψ₁}}.

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

:$B\_\{2n\}\; =\; 4n\; (-1)^\{n+1\}\; \backslash int\_0^\{\backslash infty\}\; \backslash frac\{t^\{2n-1\}\}\{e^\{2\; \backslash pi\; t\}\; -1\; \}\; \backslash mathrm\{d\}\; t$

File:BernoulliNumbersByZeta.svg

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

:{{math|1=B{{su|p=+|b=n}} = −n ζ(1 − n)}}           for {{math|n ≥ 1}} .

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

By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:{{sfnp|Arfken|1970|p=279}}

:$B\_\{2n\}\; =\; \backslash frac\; \{(-1)^\{n+1\}2(2n)!\}\; \{(2\backslash pi)^\{2n\}\}\; \backslash zeta(2n)\; \backslash quad$ for {{math|n ≥ 1}} .

Now the argument of the zeta function is positive.

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

: $|B\_\{2\; n\}|\; \backslash sim\; 4\; \backslash sqrt\{\backslash pi\; n\}\; \backslash left(\backslash frac\{n\}\{\; \backslash pi\; e\}\; \backslash right)^\{2n\}\; \backslash quad$ for {{math|n → ∞}} .

In some applications it is useful to be able to compute the Bernoulli numbers {{math|B₀}} through {{math|B_{p − 3}}} modulo {{mvar|p}}, where {{mvar|p}} is a prime; for example to test whether Vandiver's conjecture holds for {{mvar|p}}, or even just to determine whether {{mvar|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) {{math|p²}} arithmetic operations would be required. Fortunately, faster methods have been developed{{r|BuhlerCraErnMetShokrollahi2001}} which require only {{math|O(p (log p)²)}} operations (see big-O notation).

David Harvey{{r|Harvey2010}} describes an algorithm for computing Bernoulli numbers by computing {{math|B_n}} modulo {{mvar|p}} for many small primes {{mvar|p}}, and then reconstructing {{math|B_n}} via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is {{math|O(n² log(n)^{2 + ε})}} and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed {{math|B_n}} for {{math|n {{=}} 10⁸}}. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner{{r|Kellner2002}} computed {{math|B_n}} to full precision for {{math|n {{=}} 10⁶}} in December 2002 and Oleksandr Pavlyk{{r|Pavlyk29Apr2008}} for {{math|n {{=}} 10⁷}} with Mathematica in April 2008.

{{table alignment}}

:

::* Digits is to be understood as the exponent of 10 when {{math|B_n}} is written as a real number in normalized scientific notation.

Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula. 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}}

: $\backslash sum\_\{k=a\}^\{b-1\}\; f(k)\; =\; \backslash int\_a^b\; f(x)\backslash ,dx\; +\; \backslash sum\_\{k=1\}^m\; \backslash frac\{B^-\_k\}\{k!\}\; (f^\{(k-1)\}(b)-f^\{(k-1)\}(a))+R\_-(f,m).$

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

: $\backslash sum\_\{k=a+1\}^\{b\}\; f(k)\; =\; \backslash int\_a^b\; f(x)\backslash ,dx\; +\; \backslash sum\_\{k=1\}^m\; \backslash frac\{B^+\_k\}\{k!\}\; (f^\{(k-1)\}(b)-f^\{(k-1)\}(a))+R\_+(f,m).$

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

: $\backslash int\_a^b\; f(x)\backslash ,dx\; =\; f^\{(-1)\}(b)\; -\; f^\{(-1)\}(a).$

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

: $\backslash sum\_\{k=a+1\}^\{b\}\; f(k)=\; \backslash sum\_\{k=0\}^m\; \backslash frac\{B\_k\}\{k!\}\; (f^\{(k-1)\}(b)-f^\{(k-1)\}(a))+R(f,m).$

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

: $\backslash begin\{align\}$

\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{align}

Here {{math|s^{{{overline|k}}}}} denotes the rising factorial power.{{sfnp|Graham|Knuth|Patashnik|1989|loc=2.44, 2.52}}

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 {{math|ψ}}.

:$\backslash psi(z)\; \backslash sim\; \backslash ln\; z\; -\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{B^+\_k\}\{k\; z^k\}$

{{main|Faulhaber's formula}}

Bernoulli numbers feature prominently in the closed form expression of the sum of the {{math|m}}th powers of the first {{math|n}} positive integers. For {{math|m, n ≥ 0}} define

:$S\_m(n)\; =\; \backslash sum\_\{k=1\}^n\; k^m\; =\; 1^m\; +\; 2^m\; +\; \backslash cdots\; +\; n^m.$

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

: $S\_m(n)\; =\; \backslash frac\{1\}\{m\; +\; 1\}\; \backslash sum\_\{k=0\}^m\; \backslash binom\{m\; +\; 1\}\{k\}\; B^+\_k\; n^\{m\; +\; 1\; -\; k\}\; =\; m!\; \backslash sum\_\{k=0\}^m\; \backslash frac\{B^+\_k\; n^\{m\; +\; 1\; -\; k\}\}\{k!\; (m+1-k)!\}\; ,$

where {{math|({{su|p=m + 1|b=k|a=c}})}} denotes the binomial coefficient.

For example, taking {{math|m}} to be 1 gives the triangular numbers {{math|0, 1, 3, 6, ...}} {{OEIS2C|id=A000217}}.

:$1\; +\; 2\; +\; \backslash cdots\; +\; n\; =\; \backslash frac\{1\}\{2\}\; (B\_0\; n^2\; +\; 2\; B^+\_1\; n^1)\; =\; \backslash tfrac12\; (n^2\; +\; n).$

Taking {{math|m}} to be 2 gives the square pyramidal numbers {{math|0, 1, 5, 14, ...}} {{OEIS2C|id=A000330}}.

: $1^2\; +\; 2^2\; +\; \backslash cdots\; +\; n^2\; =\; \backslash frac\{1\}\{3\}\; (B\_0\; n^3\; +\; 3\; B^+\_1\; n^2\; +\; 3\; B\_2\; n^1)\; =\; \backslash tfrac13\; \backslash left(n^3\; +\; \backslash tfrac32\; n^2\; +\; \backslash tfrac12\; n\backslash right).$

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

: $S\_m(n)\; =\; \backslash frac\{1\}\{m\; +\; 1\}\; \backslash sum\_\{k=0\}^m\; (-1)^k\; \backslash binom\{m\; +\; 1\}\{k\}\; B^\{-\{\}\}\_k\; n^\{m\; +\; 1\; -\; k\}.$

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.{{r|GuoZeng2005}}

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

$$\backslash begin\{align\}$$

\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{align}

The Bernoulli numbers appear in the following Laurent series:{{sfnp|Arfken|1970|p=463}}

Digamma function: $\backslash psi(z)=\; \backslash ln\; z-\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\; \{B\_k^\{+\{\}\}\}\; \{k\; z^k\}$

The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic sphere which bound parallelizable manifolds involves Bernoulli numbers. Let {{math|ES_n}} be the number of such exotic spheres for {{Math|n ≥ 2}}, then

:$\backslash textit\{ES\}\_n\; =\; (2^\{2n-2\}-2^\{4n-3\})\; \backslash operatorname\{Numerator\}\backslash left(\backslash frac\{B\_\{4n\}\}\{4n\}\; \backslash right)\; .$

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

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^m}} is employed. The signless Worpitzky numbers are defined as

: $W\_\{n,k\}=\backslash sum\_\{v=0\}^k\; (-1)^\{v+k\}\; (v+1)^n\; \backslash frac\{k!\}\{v!(k-v)!\}\; .$

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

: $W\_\{n,k\}=k!\; \backslash left\backslash \{\; \{n+1\backslash atop\; k+1\}\; \backslash right\backslash \}.$

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

: $B\_\{n\}=\backslash sum\_\{k=0\}^n\; (-1)^k\; \backslash frac\{W\_\{n,k\}\}\{k+1\}\backslash \; =\backslash \; \backslash sum\_\{k=0\}^n\; \backslash frac\{1\}\{k+1\}\; \backslash sum\_\{v=0\}^k\; (-1)^v\; (v+1)^n\; \{k\; \backslash choose\; v\}\backslash \; .$

:{{math|1=B₀ = 1}}

:{{math|1=B₁ = 1 − {{sfrac|1|2}}}}

:{{math|1=B₂ = 1 − {{sfrac|3|2}} + {{sfrac|2|3}}}}

:{{math|1=B₃ = 1 − {{sfrac|7|2}} + {{sfrac|12|3}} − {{sfrac|6|4}}}}

:{{math|1=B₄ = 1 − {{sfrac|15|2}} + {{sfrac|50|3}} − {{sfrac|60|4}} + {{sfrac|24|5}}}}

:{{math|1=B₅ = 1 − {{sfrac|31|2}} + {{sfrac|180|3}} − {{sfrac|390|4}} + {{sfrac|360|5}} − {{sfrac|120|6}}}}

:{{math|1=B₆ = 1 − {{sfrac|63|2}} + {{sfrac|602|3}} − {{sfrac|2100|4}} + {{sfrac|3360|5}} − {{sfrac|2520|6}} + {{sfrac|720|7}}}}

This representation has {{math|B{{su|p=+|b=1}} {{=}} +{{sfrac|1|2}}}}.

Consider the sequence {{math|s_n}}, {{math|n ≥ 0}}. From Worpitzky's numbers {{OEIS2C|id=A028246}}, {{OEIS2C|id=A163626}} applied to {{math|s₀, s₀, s₁, s₀, s₁, s₂, s₀, s₁, s₂, s₃, ...}} is identical to the Akiyama–Tanigawa transform applied to {{math|s_n}} (see Connection with Stirling numbers of the first kind). This can be seen via the table:

:

The first row represents {{math|s₀, s₁, s₂, s₃, s₄}}.

Hence for the second fractional Euler numbers {{OEIS2C|id=A198631}} ({{math|n}}) / {{OEIS2C|id=A006519}} ({{math|n + 1}}):

:{{math|1= E₀ = 1}}

:{{math|1= E₁ = 1 − {{sfrac|1|2}}}}

:{{math|1= E₂ = 1 − {{sfrac|3|2}} + {{sfrac|2|4}}}}

:{{math|1= E₃ = 1 − {{sfrac|7|2}} + {{sfrac|12|4}} − {{sfrac|6|8}}}}

:{{math|1= E₄ = 1 − {{sfrac|15|2}} + {{sfrac|50|4}} − {{sfrac|60|8}} + {{sfrac|24|16}}}}

:{{math|1= E₅ = 1 − {{sfrac|31|2}} + {{sfrac|180|4}} − {{sfrac|390|8}} + {{sfrac|360|16}} − {{sfrac|120|32}}}}

:{{math|1= E₆ = 1 − {{sfrac|63|2}} + {{sfrac|602|4}} − {{sfrac|2100|8}} + {{sfrac|3360|16}} − {{sfrac|2520|32}} + {{sfrac|720|64}}}}

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for {{math|n ≥ 1}}

: $B\_n=\backslash frac\; n\; \{2^\{n+1\}-2\}\backslash sum\_\{k=0\}^\{n-1\}\; (-2)^\{-k\}\backslash ,\; W\_\{n-1,k\}\; .$

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

{{OEIS2C|id=A164555}} ({{math|n + 1}}) / {{OEIS2C|id=A027642}}({{math|n + 1}}) = {{math|{{sfrac|n + 1|2^{n + 2} − 2}}}} × {{OEIS2C|id=A198631}}({{math|n}}) / {{OEIS2C|id=A006519}}({{math|n + 1}})

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 first kind).

If one defines the Bernoulli polynomials {{math|B_k(j)}} as:{{r|Rademacher1973}}

:$B\_k(j)=k\backslash sum\_\{m=0\}^\{k-1\}\backslash binom\{j\}\{m+1\}S(k-1,m)m!+B\_k$

where {{math|B_k}} for {{math|k {{=}} 0, 1, 2,...}} are the Bernoulli numbers,

and {{math|S(k,m)}} is a Stirling number of the second kind.

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

:$B\_k(j)=\backslash sum\_\{n=0\}^k\; \backslash binom\{k\}\{n\}\; B\_n\; j^\{k-n\}.$

The coefficient of {{mvar|j}} in {{math|({{su|p=j|b=m + 1|a=c}})}} is {{math|{{sfrac|(−1)^m|m + 1}}}}.

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

: $B\_k=\backslash sum\_\{m=0\}^\{k-1\}\; (-1)^m\; \backslash frac\{m!\}\{m+1\}\; S(k-1,m)$

(resulting in {{math|B₁ {{=}} +{{sfrac|1|2}}}}) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.{{r|Boole1880|Gould1972|Apostol2010_197}}

The two main formulas relating the unsigned Stirling numbers of the first kind {{math|[{{su|p=n|b=m|a=c}}]}} to the Bernoulli numbers (with {{math|B₁ {{=}} +{{sfrac|1|2}}}}) are

: $\backslash frac\{1\}\{m!\}\backslash sum\_\{k=0\}^m\; (-1)^\{k\}\; \backslash left[\{m+1\backslash atop\; k+1\}\backslash right]\; B\_k\; =\; \backslash frac\{1\}\{m+1\},$

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

: $\backslash frac\{1\}\{m!\}\backslash sum\_\{k=0\}^m\; (-1)^k\; \backslash left[\{m+1\backslash atop\; k+1\}\backslash right]\; B\_\{n+k\}\; =\; A\_\{n,m\}.$

Here the number {{math|A_n,m}} are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.

:

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⁻ⁿ}} = 1/{{OEIS2C|id=A000079}} leads to {{OEIS2C|id=A198631}} (n) / {{OEIS2C|id=A06519}} (n + 1). Hence:

:

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^{n + 3} − 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 Worpitzky numbers).

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}}.}}

:$B^\{+\}\_n=\backslash frac$

where $|A\_n|$ is the determinant of a n-by-n Hessenberg matrix part of Pascal's triangle whose elements are: $$

a_{i, k} = \begin{cases}

0 & \text{if } k>1+i \\

{i+1 \choose k-1} & \text{otherwise}

\end{cases}

Example:

:$B^\{+\}\_6\; =\backslash frac\{\backslash det\backslash begin\{pmatrix\}$

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}

There are formulas connecting Eulerian numbers {{math|⟨{{su|p=n|b=m|a=c}}⟩}} to Bernoulli numbers:

:$\backslash begin\{align\}$

\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}

Both formulae are valid for {{math|n ≥ 0}} if {{math|B₁}} is set to {{sfrac|1|2}}. If {{math|B₁}} is set to −{{sfrac|1|2}} they are valid only for {{math|n ≥ 1}} and {{math|n ≥ 2}} respectively.

The Stirling polynomials {{math|σ_n(x)}} are related to the Bernoulli numbers by {{math|B_n {{=}} n!σ_n(1)}}. S. C. Woon described an algorithm to compute {{math|σ_n(1)}} as a binary tree:{{r|Woon1997}}

:File:SCWoonTree.png

Woon's recursive algorithm (for {{math|n ≥ 1}}) starts by assigning to the root node {{math|N {{=}} [1,2]}}. Given a node {{math|N {{=}} [a₁, a₂, ..., a_k]}} of the tree, the left child of the node is {{math|L(N) {{=}} [−a₁, a₂ + 1, a₃, ..., a_k]}} and the right child {{math|R(N) {{=}} [a₁, 2, a₂, ..., a_k]}}. A node {{math|N {{=}} [a₁, a₂, ..., a_k]}} is written as {{math|±[a₂, ..., a_k]}} in the initial part of the tree represented above with ± denoting the sign of {{math|a₁}}.

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

:$N!\; =\; a\_1\; \backslash prod\_\{k=2\}^\{\backslash operatorname\{length\}(N)\}\; a\_k!.$

Restricted to the nodes {{mvar|N}} of a fixed tree-level {{mvar|n}} the sum of {{math|{{sfrac|1|N!}}}} is {{math|σ_n(1)}}, thus

:$B\_n\; =\; \backslash sum\_\backslash stackrel\{N\; \backslash text\{\; node\; of\}\}\{\backslash text\{\; tree-level\; \}\; n\}\; \backslash frac\{n!\}\{N!\}.$

For example:

:{{math|1=B₁ = 1!({{sfrac|1|2!}})}}

:{{math|1=B₂ = 2!(−{{sfrac|1|3!}} + {{sfrac|1|2!2!}})}}

:{{math|1=B₃ = 3!({{sfrac|1|4!}} − {{sfrac|1|2!3!}} − {{sfrac|1|3!2!}} + {{sfrac|1|2!2!2!}})}}

The integral

: $b(s)\; =\; 2e^\{s\; i\; \backslash pi/2\}\backslash int\_0^\backslash infty\; \backslash frac\{st^s\}\{1-e^\{2\backslash pi\; t\}\}\; \backslash frac\{dt\}\{t\}\; =\; \backslash frac\{s!\}\{2^\{s-1\}\}\backslash frac\{\backslash zeta(s)\}\{\{\; \}\backslash pi^s\{\; \}\}(-i)^s=\; \backslash frac\{2s!\backslash zeta(s)\}\{(2\backslash pi\; i)^s\}$

has as special values {{math|b(2n) {{=}} B_2n}} for {{math|n > 0}}.

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

: $\backslash begin\{align\}$

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}

Another similar integral representation is

: $b(s)\; =\; -\backslash frac\{e^\{s\; i\; \backslash pi/2\}\}\{2^\{s\}-1\}\backslash int\_0^\backslash infty\; \backslash frac\{st^\{s\}\}\{\backslash sinh\backslash pi\; t\}\; \backslash frac\{dt\}\{t\}=\; \backslash frac\{2e^\{s\; i\; \backslash pi/2\}\}\{2^\{s\}-1\}\backslash int\_0^\backslash infty\; \backslash frac\{e^\{\backslash pi\; t\}st^s\}\{1-e^\{2\backslash pi\; t\}\}\; \backslash frac\{dt\}\{t\}.$

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 {{math|E_2n}} are in magnitude approximately {{math|{{sfrac|2|π}}(4²ⁿ − 2²ⁿ)}} times larger than the Bernoulli numbers {{math|B_2n}}. In consequence:

: $\backslash pi\; \backslash sim\; 2\; (2^\{2n\}\; -\; 4^\{2n\})\; \backslash frac\{B\_\{2n\}\}\{E\_\{2n\}\}.$

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.

Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd {{mvar|n}}, {{math|B_n {{=}} E_n {{=}} 0}} (with the exception {{math|B₁}}), it suffices to consider the case when {{mvar|n}} is even.

:$\backslash begin\{align\}$

B_n &= \sum_{k=0}^{n-1}\binom{n-1}{k} \frac{n}{4^n-2^n}E_k & n&=2, 4, 6, \ldots \\[6pt]

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}

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{{citation

| 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 }}{{r|Elkies2003}}

:$S\_n\; =\; 2\; \backslash left(\backslash frac\{2\}\{\backslash pi\}\backslash right)^n\; \backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash frac\{\; (-1)^\{kn\}\; \}\{(2k+1)^n\}\; =\; 2\; \backslash left(\backslash frac\{2\}\{\backslash pi\}\backslash right)^n\; \backslash lim\_\{K\backslash to\; \backslash infty\}\; \backslash sum\_\{k\; =\; -K\}^K\; (4k+1)^\{-n\}.$

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.{{r|Euler1735}} The first few of these numbers are

: $S\_n\; =\; 1,1,\backslash frac\{1\}\{2\},\backslash frac\{1\}\{3\},\backslash frac\{5\}\{24\},\; \backslash frac\{2\}\{15\},\backslash frac\{61\}\{720\},\backslash frac\{17\}\{315\},\backslash frac\{277\}\{8064\},\backslash frac\{62\}\{2835\},\backslash ldots$ ({{OEIS2C|id=A099612}} / {{OEIS2C|id=A099617}})

These are the coefficients in the expansion of {{math|sec x + tan x}}.

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

: $\backslash begin\{align\}$

B_{n} &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] \frac{n! }{2^n - 4^n}\, S_{n}\ , & n&= 2, 3, \ldots \\

E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] n! \, S_{n+1} & n &= 0, 1, \ldots

\end{align}

The expression [{{math|n}} even] has the value 1 if {{math|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 {{math|R_n {{=}} {{sfrac|2S_n|S_{n + 1}}}}} when {{mvar|n}} is even. The {{math|R_n}} 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}}):

: $2,\; 4,\; 3,\; \backslash frac\{16\}\{5\},\; \backslash frac\{25\}\{8\},\; \backslash frac\{192\}\{61\},\; \backslash frac\{427\}\{136\},\; \backslash frac\{4352\}\{1385\},\; \backslash frac\{12465\}\{3968\},\; \backslash frac\{158720\}\{50521\},\backslash ldots\; \backslash quad\; \backslash longrightarrow\; \backslash pi.$

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

Consider the Akiyama–Tanigawa transform for the sequence {{OEIS2C|id=A046978}} ({{math|n + 2}}) / {{OEIS2C|id=A016116}} ({{math|n + 1}}):

:

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}}.

The sequence S_n has another unexpected yet important property: The denominators of S_n+1 divide the factorial {{math|n!}}. In other words: the numbers {{math|1=T_n = S_{n + 1} n!}}, sometimes called Euler zigzag numbers, are integers.

: $T\_n\; =\; 1,\backslash ,1,\backslash ,1,\backslash ,2,\backslash ,5,\backslash ,16,\backslash ,61,\backslash ,272,\backslash ,1385,\backslash ,7936,\backslash ,50521,\backslash ,353792,\backslash ldots\; \backslash quad\; n=0,\; 1,\; 2,\; 3,\; \backslash ldots$ ({{OEIS2C|id=A000111}}). See ({{OEIS2C|id=A253671}}).

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

: $\backslash sum\_\{n=0\}^\backslash infty\; T\_n\; \backslash frac\{x^n\}\{n!\}\; =\; \backslash tan\; \backslash left(\backslash frac\backslash pi4\; +\; \backslash frac\; x2\backslash right)\; =\; \backslash sec\; x\; +\; \backslash tan\; x$.

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

:$\backslash begin\{align\}$

B_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [n\text{ even}] \frac{n }{2^n-4^n}\, T_{n-1}\ & n &\geq 2 \\

E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [n\text{ even}] T_{n} & n &\geq 0

\end{align}

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers {{math|E_2n}} are given immediately by {{math|T_2n}} and the Bernoulli numbers {{math|B_2n}} are fractions obtained from {{math|T_{2n - 1}}} by some easy shifting, avoiding rational arithmetic.

What remains is to find a convenient way to compute the numbers {{math|T_n}}. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate {{math|T_n}}.{{r|Seidel1877}}

{{image frame|align=none|caption=Seidel's algorithm for {{math|T_n}}

|content=$$

\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}

}}

1. Start by putting 1 in row 0 and let {{math|k}} denote the number of the row currently being filled
2. 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
3. At the end of the row duplicate the last number.
4. 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.

Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers {{math|T_2n}} and recommended this method for computing {{math|B_2n}} and {{math|E_2n}} 'on electronic computers using only simple operations on integers'.{{r|KnuthBuckholtz1967}}

V. I. Arnold{{r|Arnold1991}} rediscovered Seidel's algorithm and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.

Triangular form:

:

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:

:

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:

:

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

:

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:

:

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}}) = 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}}...}}.

:

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:

:

The first two upper diagonals are {{nowrap|−1 3 −24 402...}} = {{math|(−1)^{n + 1}}} × {{OEIS2C|id=A002832}}. The sum of the antidiagonals is {{nowrap|0 −2 0 10...}} = 2 × {{OEIS2C|id=A122045}}(n + 1).

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

:

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 × {{OEIS2C|id=A155585}}({{math|n + }}1). {{OEIS2C|id=A163747}} − {{OEIS2C|id=A163982}} = 2 × {{OEIS2C|id=A122045}}.

{{main|Alternating permutations}}

Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis.{{r|André1879|André1881}} Looking at the first terms of the Taylor expansion of the trigonometric functions

{{math|tan x}} and {{math|sec x}} André made a startling discovery.

:$\backslash begin\{align\}$

\tan x &= x + \frac{2x^3}{3!} + \frac{16x^5}{5!} + \frac{272x^7}{7!} + \frac{7936x^9}{9!} + \cdots\\[6pt]

\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}

The coefficients are the Euler numbers 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_n}}.

: $\backslash tan\; x\; +\; \backslash sec\; x\; =\; 1\; +\; x\; +\; \backslash tfrac\{1\}\{2\}x^2\; +\; \backslash tfrac\{1\}\{3\}x^3\; +\; \backslash tfrac\{5\}\{24\}x^4\; +\; \backslash tfrac\{2\}\{15\}x^5\; +\; \backslash tfrac\{61\}\{720\}x^6\; +\; \backslash cdots$

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).

The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers:

{{math|1=B₀ = 1}}, {{math|1=B₁ = 0}}, {{math|1=B₂ = {{sfrac|1|6}}}}, {{math|1=B₃ = 0}}, {{math|1=B₄ = −{{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|B₁}}, named intrinsic Bernoulli numbers {{math|B_i(n)}}.

:

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:

:

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_i(n) {{=}} 1, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, 0, −{{sfrac|17|8}}, 0, ...}} The corresponding Akiyama transform is:

:

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:

:

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as {{math|B_n {{=}} −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|B₁ {{=}} {{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 Agoh–Giuga conjecture postulates that {{mvar|p}} is a prime number if and only if {{math|pB_{p − 1}}} is congruent to −1 modulo {{mvar|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 Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem,{{r|Kummer1850}} which says:

:If the odd prime {{mvar|p}} does not divide any of the numerators of the Bernoulli numbers {{math|B₂, B₄, ..., B_{p − 3}}} then {{math|x^p + y^p + z^p {{=}} 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.{{r|Kummer1851}}

{{main|Kummer's congruence}}

:Let {{mvar|p}} be an odd prime and {{mvar|b}} an even number such that {{math|p − 1}} does not divide {{mvar|b}}. Then for any non-negative integer {{mvar|k}}

:: $\backslash frac\{B\_\{k(p-1)+b\}\}\{k(p-1)+b\}\; \backslash equiv\; \backslash frac\{B\_\{b\}\}\{b\}\; \backslash pmod\{p\}.$

A generalization of these congruences goes by the name of {{math|p}}-adic continuity.

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^{b − 1} (p − 1))}}, then

:$(1-p^\{m-1\})\backslash frac\{B\_m\}\{m\}\; \backslash equiv\; (1-p^\{n-1\})\backslash frac\{B\_n\}\; n\; \backslash pmod\{p^b\}.$

Since {{math|B_n {{=}} −nζ(1 − n)}}, this can also be written

:$\backslash left(1-p^\{-u\}\backslash right)\backslash zeta(u)\; \backslash equiv\; \backslash left(1-p^\{-v\}\backslash right)\backslash zeta(v)\; \backslash pmod\{p^b\},$

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^−s}} taken out of the Euler product formula, is continuous in the p-adic numbers 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_p(s)}} for all {{mvar|p}}-adic integers $\backslash mathbb\{Z\}\_p,$ the p-adic zeta function.

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:

:$\backslash binom\{m+3\}\{m\}\; B\_m=\backslash begin\{cases\}$

\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}

{{main|Von Staudt–Clausen theorem}}

The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt{{r|vonStaudt1840}} and Thomas Clausen{{r|Clausen1840}} independently in 1840. The theorem states that for every {{math|n > 0}},

: $B\_\{2n\}\; +\; \backslash sum\_\{(p-1)\backslash ,\backslash mid\backslash ,2n\}\; \backslash frac1p$

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

A consequence of this is that the denominator of {{math|B_2n}} is given by the product of all primes {{math|p}} for which {{math|p − 1}} divides {{math|2n}}. In particular, these denominators are square-free and divisible by 6.

The sum

:$\backslash varphi\_k(n)\; =\; \backslash sum\_\{i=0\}^n\; i^k\; -\; \backslash frac\{n^k\}\; 2$

can be evaluated for negative values of the index {{math|n}}. Doing so will show that it is an odd function 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_{2k + 1 − m}}} is 0 for {{math|m}} even and {{math|2k + 1 − m > 1}}; and that the term for {{math|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 {{math|n > 1}} the number {{math|2B_n}} 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

: $2B\_n\; =\backslash sum\_\{m=0\}^n\; (-1)^m\; \backslash frac\{2\}\{m+1\}m!\; \backslash left\backslash \{\{n+1\backslash atop\; m+1\}\; \backslash right\backslash \}\; =\; 0\backslash quad(n>1\; \backslash text\{\; is\; odd\})$

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 {{math|S_n,m}} be the number of surjective maps from {{math|1={1, 2, ..., n}}} to {{math|1={1, 2, ..., m}}}, then {{math|S_n,m {{=}} m!{{{su|p=n|b=m|a=c}}}}}. The last equation can only hold if

: $\backslash sum\_\{\backslash text\{odd\; \}m=1\}^\{n-1\}\; \backslash frac\; 2\; \{m^2\}S\_\{n,m\}=\backslash sum\_\{\backslash text\{even\; \}\; m=2\}^n\; \backslash frac\{2\}\{m^2\}\; S\_\{n,m\}\; \backslash quad\; (n>2\; \backslash text\{\; is\; even\}).$

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

:{{math|1=n = 4: 2 + 8 = 7 + 3}},

:{{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.

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:{{r|Riesz1916}}

:For every {{math|ε > {{sfrac|1|4}}}} there exists a constant {{math|C_ε > 0}} (depending on {{math|ε}}) such that {{math|{{abs|R(x)}} < C_εx^ε}} as {{math|x → ∞}}.

Here {{math|R(x)}} is the Riesz function

: $R(x)\; =\; 2\; \backslash sum\_\{k=1\}^\backslash infty$

\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}}.

{{math|n^{{{overline|k}}}}} denotes the rising factorial power in the notation of D. E. Knuth. The numbers {{math|β_n {{=}} {{sfrac|B_n|n}}}} occur frequently in the study of the zeta function and are significant because {{math|β_n}} is a {{math|p}}-integer for primes {{math|p}} where {{math|p − 1}} does not divide {{math|n}}. The {{math|β_n}} are called divided 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-function in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.

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

: $\backslash sum\_\{a=1\}^f\; \backslash chi(a)\; \backslash frac\{te^\{at\}\}\{e^\{ft\}-1\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; B\_\{k,\backslash chi\}\backslash frac\{t^k\}\{k!\}.$

Apart from the exceptional {{math|B_1,1 {{=}} {{sfrac|1|2}}}}, we have, for any Dirichlet character {{mvar|χ}}, that {{math|B_k,χ {{=}} 0}} if {{math|χ(−1) ≠ (−1)^k}}.

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}}:

: $L(1-k,\backslash chi)=-\backslash frac\{B\_\{k,\backslash chi\}\}k,$

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

{{main|Eisenstein–Kronecker number}}

Eisenstein–Kronecker numbers are an analogue of the generalized Bernoulli numbers for imaginary quadratic fields.{{r|Charollois-Sczech|BK}} They are related to critical L-values of Hecke characters.{{r|BK}}

{{unordered list

|1 = Umbral calculus gives a compact form of Bernoulli's formula by using an abstract symbol {{math|B}}:

: $S\_m(n)\; =\; \backslash frac\; 1\; \{m+1\}\; ((\backslash mathbf\{B\}\; +\; n)^\{m+1\}\; -\; B\_\{m+1\})$

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

:$S\_m(n)\; =\; \backslash int\_0^n\; (\backslash mathbf\{B\}+x)^m\backslash ,dx$

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

:$(1-2\backslash mathbf\{B\})^m\; =\; (2-2^m)\; B\_m$

|2 = Let {{math|n}} be non-negative and even

:$\backslash zeta(n)\; =\; \backslash frac\{(-1)^\{\backslash frac\{n\}\{2\}\; -\; 1\}\; B\_n\; (2\backslash pi)^n\}\{2(n!)\}$

|3 = The {{math|n}}th cumulant of the uniform probability distribution on the interval [−1, 0] is {{math|{{sfrac|B_n|n}}}}.

|4 = Let {{math|n? {{=}} {{sfrac|1|n!}}}} and {{math|n ≥ 1}}. Then {{math|B_n}} is the following {{math|(n + 1) × (n + 1)}} determinant:{{r|Malenfant2011}}

: $$

\begin{align}

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} \\[8pt]

& = 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}

Thus the determinant is {{math|σ_n(1)}}, the Stirling polynomial at {{math|x {{=}} 1}}.

|5 = For even-numbered Bernoulli numbers, {{math|B_2p}} is given by the {{math|(p + 1) × (p + 1)}} determinant::{{r|Malenfant2011}}

:$B\_\{2p\}\; =\; -\backslash frac\{(2p)!\}\{2^\{2p\}\; -\; 2\}\; \backslash begin\{vmatrix\}$

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}

|6 = Let {{math|n ≥ 1}}. Then (Leonhard Euler)Euler, E41, Inventio summae cuiusque seriei ex dato termino generali

: $\backslash frac\{1\}\{n\}\; \backslash sum\_\{k=1\}^n\; \backslash binom\{n\}\{k\}B\_k\; B\_\{n-k\}+B\_\{n-1\}=-B\_n$

|7 = Let {{math|n ≥ 1}}. Then{{r|vonEttingshausen1827}}

: $\backslash sum\_\{k=0\}^n\; \backslash binom\{n+1\}k\; (n+k+1)B\_\{n+k\}=0$

|8 = Let {{math|n ≥ 0}}. Then (Leopold Kronecker 1883)

: $B\_n\; =\; -\; \backslash sum\_\{k=1\}^\{n+1\}\; \backslash frac\{(-1)^k\}\{k\}\; \backslash binom\{n+1\}\{k\}\; \backslash sum\_\{j=1\}^k\; j^n$

|9 = Let {{math|n ≥ 1}} and {{math|m ≥ 1}}. Then{{r|Carlitz1968}}

: $(-1)^m\; \backslash sum\_\{r=0\}^m\; \backslash binom\{m\}\{r\}\; B\_\{n+r\}=(-1)^n\; \backslash sum\_\{s=0\}^n\; \backslash binom\{n\}\{s\}\; B\_\{m+s\}$

|10 = Let {{math|n ≥ 4}} and

: $H\_n=\backslash sum\_\{k=1\}^n\; k^\{-1\}$

the harmonic number. Then (H. Miki 1978)

: $\backslash frac\{n\}\{2\}\backslash sum\_\{k=2\}^\{n-2\}\backslash frac\{B\_\{n-k\}\}\{n-k\}\backslash frac\{B\_k\}\{k\}\; -\; \backslash sum\_\{k=2\}^\{n-2\}\; \backslash binom\{n\}\{k\}\backslash frac\{B\_\{n-k\}\}\{n-k\}\; B\_k\; =H\_n\; B\_n$

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

: $(n+2)\backslash sum\_\{k=2\}^\{n-2\}B\_k\; B\_\{n-k\}-2\backslash sum\_\{l=2\}^\{n-2\}\backslash binom\{n+2\}\{l\}\; B\_l\; B\_\{n-l\}=n(n+1)B\_n$

|12 = Faber–Pandharipande–Zagier–Gessel identity: for {{math|n ≥ 1}},

: $\backslash frac\{n\}\{2\}\backslash left(B\_\{n-1\}(x)+\backslash sum\_\{k=1\}^\{n-1\}\backslash frac\{B\_\{k\}(x)\}\{k\}$

\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 {{math|x {{=}} 0}} or {{math|x {{=}} 1}} results in the Bernoulli number identity in one or another convention.

|13 = The next formula is true for {{math|n ≥ 0}} if {{math|B₁ {{=}} B₁(1) {{=}} {{sfrac|1|2}}}}, but only for {{math|n ≥ 1}} if {{math|B₁ {{=}} B₁(0) {{=}} −{{sfrac|1|2}}}}.

:$\backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash frac\{B\_k\}\{n-k+2\}\; =\; \backslash frac\{B\_\{n+1\}\}\{n+1\}$

|14 = Let {{math|n ≥ 0}}. Then

:$-1\; +\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash frac\{2^\{n-k+1\}\}\{n-k+1\}B\_k(1)\; =\; 2^n$

and

:$-1\; +\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash frac\{2^\{n-k+1\}\}\{n-k+1\}B\_\{k\}(0)\; =\; \backslash delta\_\{n,0\}$

|15 = A reciprocity relation of M. B. Gelfand:{{r|AgohDilcher2008}}

: $(-1)^\{m+1\}\; \backslash sum\_\{j=0\}^k\; \backslash binom\{k\}\{j\}\; \backslash frac\{B\_\{m+1+j\}\}\{m+1+j\}\; +\; (-1)^\{k+1\}\; \backslash sum\_\{j=0\}^m\; \backslash binom\{m\}\{j\}\backslash frac\{B\_\{k+1+j\}\}\{k+1+j\}\; =\; \backslash frac\{k!m!\}\{(k+m+1)!\}$

}}

{{div col|colwidth=20em}}

• Bernoulli polynomial
• Bernoulli polynomials of the second kind
• Bernoulli umbra
• Bell number
• Euler number
• Genocchi number
• Kummer's congruences
• Poly-Bernoulli number
• Hurwitz zeta function
• Euler summation
• Stirling polynomial
• Sums of powers

{{div col end}}

{{Notelist}}

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{{Neukirch ANT|mode=cs2}} §VII.2.

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