Bernoulli number Information & Bernoulli number Links at HealthHaven.com

In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers.

B_n = 0 for all odd n other than 1. B₁ = 1/2 or −1/2 depending on the convention adopted. The values of the first few nonzero Bernoulli numbers are (more values below):

n	0	1	2	4	6	8	10	12
B_n	1	±1/2	1/6	-1/30	1/42	-1/30	5/66	-691/2730

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712^[1]^[2] in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713.

They appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

In note G of Ada Lovelace's notes on the analytical engine from 1842, Lovelace describes an algorithm for generating Bernoulli numbers with Babbage's machine^[3]. As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.

1 Sum of powers
2 Definitions
- 2.1 Recursive definition
- 2.2 Explicit definition
- 2.3 Generating function
- 2.4 Algorithmic description
3 Efficient computation of Bernoulli numbers
4 Different viewpoints and conventions
5 Applications of the Bernoulli numbers
- 5.1 Asymptotic analysis
- 5.2 Use in topology
6 Combinatorial definitions
- 6.1 Connection with the Worpitzky number
- 6.2 Connection with the Stirling set number
- 6.3 Connection with the Stirling cycle number
- 6.4 Connection with the Eulerian number
7 A binary tree representation
8 Asymptotic approximation
9 Integral representation and continuation
10 The relation to the Euler numbers and π
11 An algorithmic view: the Seidel triangle
12 A combinatorial view: alternating permutations
13 Generalizations by polynomials
14 Arithmetical properties of the Bernoulli numbers
- 14.1 The Kummer theorems
- 14.2 p-adic continuity
- 14.3 Ramanujan's congruences
- 14.4 Von Staudt–Clausen theorem
- 14.5 Why do the odd Bernoulli numbers vanish?
- 14.6 A restatement of the Riemann hypothesis
15 History
- 15.1 Early history
- 15.2 Reconstruction of 'Summae Potestatum'
16 Appendix
- 16.1 Assorted identities
- 16.2 Values of the Bernoulli numbers
17 See also
18 Notes
19 References
20 External links

[edit] Sum of powers

Main article: Faulhaber's formula

Closed forms of the sum of powers for fixed values of m

$S_m(n) = \sum_{k=1}^{n} k^m = 1^m + 2^m + \cdots + {n}^m \,$

are always polynomials in n of degree m + 1. Note that S_m(0) = 0 for all m ≥ 0 because in this case the sum is the empty sum. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:

$S_m(n) = {1\over{m+1}}\sum_{k=0}^m{m+1\choose{k}} B_k n^{m+1-k}$

Let n ≥ 0. Taking m to be 0 and B₀ = 1 gives the natural number 0,1,2,3,… (sequence A001477).

$0 + 1 + 1 + \cdots + 1 = \frac{1}{1}\left(B_0 n\right) = n.$

Taking m to be 1 and B₁ = 1/2 gives the triangular number 0,1,3,6,… (sequence A000217).

$0 + 1 + 2 + \cdots + n = \frac{1}{2}\left(B_0 n^2+2B_1 n^1\right) = \frac{1}{2}\left(n^2+n\right).$

Taking m to be 2 and B₂ = 1/6 gives the square pyramidal number 0,1,5,14,… (sequence A000330).

$0 + 1^2 + 2^2 + \cdots + n^2 = \frac{1}{3}\left(B_0 n^3+3B_1 n^2+3B_2 n^1 \right) = \frac{1}{3}\left(n^3+\frac{3}{2}n^2+\frac{1}{2}n\right).$

Although Bernoulli's formula reflects faithfully what Bernoulli has written some authors state Bernoulli's formula in a different way which is neither in accordance with Bernoulli's statement nor has any obvious advantage compared to it. They write:

$S_m(n) = {1\over{m+1}}\sum_{k=0}^m (-1)^k {m+1\choose{k}} B_k n^{m+1-k}$

To avoid a contradiction to the valid formula above these authors have to set B₁ = −1/2. In the next section consequences of the resulting differences will be commented on as they are likely to produce some confusion.

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

Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).

[edit] 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 algorithmic description.

For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like (Ireland & Rosen 1990) or (Conway & Guy 1996).

Unfortunately in the literature the definition is given in two variants: Despite the fact, that Bernoulli defined B₁ = 1/2 some authors set B₁ = —1/2 (more on different conventions below). In order to prevent potential confusions both variants will be described here, side by side.

[edit] Recursive definition

The recursive equation is best introduced in a slightly more general form

$B_m(n)=n^m-\sum_{k=0}^{m-1}\binom mk\frac{B_k(n)}{m-k+1} \ .$

This equation defines rational numbers B_m(n) for all integers n ≥ 0, m ≥ 0. 0⁰ has to be interpreted as 1. The recursion has its base in B₀(n) = 1 for all n. The two variants now follow by setting n = 0 respectively n = 1. Additionally the notation is simplified by erasing the reference to the parameter n.

n = 0	n = 1
$B_m = \left[ m = 0 \right] -\sum_{k=0}^{m-1}\binom mk\frac{B_k}{m-k+1}$	$B_m= 1 - \sum_{k=0}^{m-1}\binom mk\frac{B_k}{m-k+1}$

Here the expression [m = 0] has the value 1 if m = 0 and 0 otherwise (Iverson bracket). Whenever a confusion between the two kinds of definitions might arise it can be avoided by referring to the more general definition and by reintroducing the erased parameter: writing B_m(0) in the first case and B_m(1) in the second will unambiguously denote the value in question.

[edit] Explicit definition

Starting again with a slightly more general formula

$B_m(n)=\sum_{k=0}^m\sum_{v=0}^k(-1)^v\binom kv\frac{\left( n+v\right) ^m}{k+1} ,$

the choices n = 0 and n = 1 lead to

n = 0	n = 1
$B_m=\sum_{k=0}^m\sum_{v=0}^k(-1)^v\binom kv\frac{v^m}{k+1} \ ,$	$B_m=\sum_{k=1}^{m+1}\sum_{v=1}^{k+1}(-1)^{v+1}\binom{k-1}{v-1}\frac{v^m}k \ .$

There is a widespread misinformation that no simple closed formulas for the Bernoulli numbers exist. The last two equations show that this is not true. Moreover, already in 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers (Saalschütz 1893), usually giving some reference in the older literature.

[edit] Generating function

The general formula for the generating function is

$\frac{te^{nt}}{e^t-1}=\sum_{m=0}^\infty B_m(n)\frac{t^m}{m!} \ .$

The choices n = 0 and n = 1 lead to

n = 0	n = 1
$\frac t{e^t-1}=\sum_{m=0}^\infty B_m\frac{t^m}{m!} \ ,$	$\frac t{1-e^{-t}}=\sum_{m=0}^\infty B_m\frac{t^m}{m!} \ .$

[edit] Algorithmic description

Although the above recursive formula can be used for computation it is mainly used to establish the connection with the sum of powers because it is computationally expensive. However, both simple and high-end algorithms for computing Bernoulli numbers exist. A simple one is given in pseudocode below in the text box 'Akiyama–Tanigawa algorithm' and pointers to high-end algorithms are given the next section below.

[edit] Efficient computation of Bernoulli numbers

In some applications it is useful to be able to compute the Bernoulli numbers B₀ through B_p − 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) p² arithmetic operations would be required. Fortunately, faster methods have been developed (Buhler et al. 2001) which require only O(p (log p)²) operations (see big-O notation).

David Harvey (Harvey 2008) describes an algorithm for computing Bernoulli numbers by computing B_n modulo p for many small primes p, and then reconstructing B_n via the Chinese Remainder Theorem. Harvey writes that the asymptotic time complexity of this algorithm is O(n² log(n)^2+eps) and claims that this implementation is significantly faster than implementations based on other methods. Harvey's implementation is included in Sage since version 3.1. Using this implementation Harvey computed B_n for n = 10⁸, which is a new record (October 2008). Prior to that Bernd Kellner (Kellner 2002) computed B_n to full precision for n = 10⁶ in December 2002 and Oleksandr Pavlyk (Pavlyk 2008) for n = 10⁷ with 'Mathematica' in April 2008.

History of the computation of Bernoulli numbers
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`	`360`	`478`
`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 B(n) is written as a real in normalized scientific notation.

[edit] Different viewpoints and conventions

The Bernoulli numbers can be regarded from four main viewpoints:

as standalone arithmetical objects,
as combinatorial objects,
as values of a sequence of certain polynomials,
as values of the Riemann zeta function.

Each of these viewpoints leads to a set of more or less different conventions.

Bernoulli numbers as standalone arithmetical objects.
Associated sequence: 1/6, −1/30, 1/42, −1/30,...
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 'archaic'. For example Jean-Pierre Serre uses it in his highly acclaimed book A Course in Arithmetic which is a standard textbook used at many universities today.
Bernoulli numbers as combinatorial objects.
Associated sequence: 1, +1/2, 1/6, 0,....
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_n(x), formula (6.52) in Concrete Mathematics by Graham, Knuth and Patashnik.

$\left(\frac{ze^{z}}{e^{z}-1} \right)^{x} = x\sum_{n\geq0}\sigma_{n}(x)z^{n}$

In consequence B_n = n! σ_n(1) for n ≥ 0.
Bernoulli numbers as values of a sequence of certain polynomials.
Assuming the Bernoulli polynomials as already introduced the Bernoulli numbers can be defined in two different ways:
B_n = B_n(0). Associated sequence: 1, −1/2, 1/6, 0,....
B_n = B_n(1). Associated sequence: 1, +1/2, 1/6, 0,....
The two definitions differ only in the sign of B₁. The choice B_n = B_n(0) is the convention used in the Handbook of Mathematical Functions.
Bernoulli numbers as values of the Riemann zeta function.
Associated sequence: 1, +1/2, 1/6, 0,....

The Bernoulli numbers as given by the Riemann zeta function.

This convention agrees with the convention B_n = B_n(1) (for example J. Neukirch and M. Kaneko). The sign '+' for B₁ matches the representation of the Bernoulli numbers by the Riemann zeta function. In fact the identity nζ(1 − n) = (−1)ⁿ⁺¹B_n valid for all n > 0 is then replaced by the simpler nζ(1 − n) = −B_n. (See the paper of S. C. Woon.)
(Note that in the foregoing equation for n = 0 and n = 1 the expression −nζ(1 − n) is to be understood as lim_x → n −xζ(1 − x).)

$\ B_n = n!\sigma_{n}(1) = B_n(1) = -n\zeta(1-n) \quad (n \geq 0) \$

[edit] Applications of the Bernoulli numbers

[edit] Asymptotic analysis

Arguably the most important application of the Bernoulli number in mathematics is their use in the Euler–MacLaurin formula. Assuming that ƒ is a sufficiently often differentiable function the Euler–MacLaurin formula can be written as ^[4]

$\sum\limits_{a\leq k<b}f(k)=\int_a^b f(x)\,dx \ + \sum\limits_{k=1}^m \frac{B_k}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R(f,m) \ .$

This formulation assumes the convention B₁ = −1/2. However, if one sets B₁ = 1/2 then this formula can also be written as

$\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). \$

Here ƒ⁽⁰⁾ = ƒ which is a commonly used notation identifying the zero-th derivative of ƒ with ƒ. Moreover, let ƒ⁽⁻¹⁾ denote an antiderivative of ƒ. By the fundamental theorem of calculus,

$\int_a^b f(x)\,dx\ = f^{(-1)}(b) - f^{(-1)}(a).$

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

$\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). \$

This form is for example the source for the important Euler–MacLaurin expansion of the zeta function (B₁ = 1/2)

$\begin{align} \zeta(s) & =\sum_{k=0}^m \frac{B_k}{k!} s^{\overline{k-1}} + R(s,m) \\ & = \frac{B_0}{0!}s^{\overline{-1}} + \frac{B_1}{1!} s^{\overline{0}} + \frac{B_2}{2!} s^{\overline{1}} +\cdots+R(s,m) \\ & =\frac{1}{s-1} + \frac{1}{2} + \frac{1}{12}s + \cdots + R(s,m). \end{align}$

Here $s^{\overline{k}}$ denotes the rising factorial power.^[5]

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 (again B₁ = 1/2).

$\psi(z) \sim \ln z - \sum_{k=1}^{\infty} \frac{B_{k}}{k z^k}$

[edit] 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 ES_n be the number of such exotic spheres for n ≥ 2, then (see also sequence A047680)

$ES_n = (2^{2n-2}-2^{4n-3}) \ \text{Numerator} \left(\frac{B_{4n}}{4n} \right) .$

[edit] Combinatorial definitions

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.

[edit] Connection with the Worpitzky number

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

$W_{n,k}=\sum_{v=0}^{k}(-1)^{v+k} \left(v+1\right)^{n} \frac{k!}{v!(k-v)!} \ .$

They can also be expressed through the Stirling set number

$W_{n,k}=k! \left\{\begin{matrix} n+1 \\ k+1 \end{matrix}\right\}.$

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

$B_{n}=\sum_{k=0}^{n}(-1)^{k}\frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^{n}\frac{1}{k+1}\sum_{v=0}^{k}(-1)^v {k \choose v} \left(v+1\right)^{n} \ .$

This representation has B₁ = 1/2.

Worpitzky's representation of the Bernoulli number
B₀	=	1/1
B₁	=	1/1 − 1/2
B₂	=	1/1 − 3/2 + 2/3
B₃	=	1/1 − 7/2 + 12/3 − 6/4
B₄	=	1/1 − 15/2 + 50/3 − 60/4 + 24/5
B₅	=	1/1 − 31/2 + 180/3 − 390/4 + 360/5 − 120/6
B₆	=	1/1 − 63/2 + 602/3 − 2100/4 + 3360/5 − 2520/6 + 720/7

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

$B_{n}=\frac{n}{2^{n+1}-2}\sum_{k=0}^{n-1} (-2)^{-k}\; W_{n-1,k} \ .$

[edit] Connection with the Stirling set number

A similar combinatorial representation derives from

$B_{n}=\sum_{k=0}^{n}(-1)^{k+n}\frac{k!}{k+1} \left\{\begin{matrix} n \\ k \end{matrix}\right\} \ .$

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 + 1. The combinatorics of this representation can be seen from:

$B_{0}= +\left\vert \ \left\{ \varnothing \right\} \ \right\vert$

$B_{1}= -\frac{1}{1} \left\vert \ \varnothing \ \right\vert + \frac{1}{2}\left\vert \left\{a\right\}\right\vert$

$B_{2}= +\frac{1}{1} \left\vert \ \varnothing \ \right\vert - \frac{1}{2}\left\vert \left\{aa\right\}\right\vert +\frac{1}{3}\left\vert \left\{ab,ba\right\} \right\vert$

$B_{3}= -\frac{1}{1} \left\vert \ \varnothing \ \right\vert +\frac{1}{2}\left\vert \left\{aaa\right\}\right\vert -\frac{1}{3}\left\vert \left\{aab,aba,baa,abb,bab,bba\right\}\right\vert +\frac{1}{4}\left\vert \left\{abc,acb,bac,bca,cab,cba\right\} \right\vert$

[edit] Connection with the Stirling cycle number

Let $\left[\begin{matrix} n \\ m \end{matrix}\right]$ denote the signless Stirling cycle number. The two main formulas relating these number to the Bernoulli number (B₁ = 1/2) are

$\frac{1}{m!}\sum_{k=0}^{m}(-1)^{k} \left[\begin{matrix} m+1 \\ k+1 \end{matrix}\right] B_{k}=\frac{1}{m+1}\ ,$

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

$\frac{1}{m!}\sum_{k=0}^{m}(-1)^{k} \left[\begin{matrix} m+1 \\ k+1 \end{matrix}\right] B_{n+k} = A_{n,m} \ .$

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

Akiyama–Tanigawa number
n \ m	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	...	...	...	...

These relations lead to a simple algorithm to compute the Bernoulli number. The input is the first row, A_0,m = 1/(m + 1) and the output are the Bernoulli number in the first column A_n,0 = B_n . This transformation is shown in pseudo-code below.

Akiyama–Tanigawa algorithm for B_n

Enter integer n.

For m from 0 by 1 to n do

A[m] ← 1/(m + 1)

For j from m by −1 to 1 do

A[j − 1] ← j × (A[j − 1] − A[j])

Return A[0] (which is B_n).

[edit] Connection with the Eulerian number

The Eulerian number 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 ^[6] written as $\left \langle {n\atop m} \right \rangle .$ The two main formulas connecting the Eulerian number to the Bernoulli number are:

$\sum_{m=0}^{n}(-1)^{m} {\left \langle {n\atop m} \right \rangle} = 2^{n+1}(2^{n+1}-1) \frac{B_{n+1}}{n+1} \ ,$

$\sum_{m=0}^{n}(-1)^{m} {\left \langle {n\atop m} \right \rangle} {\binom{n}{m}}^{-1} = (n+1) B_{n} \ .$

Both formulas 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.

[edit] A binary tree representation

The Stirling polynomials σ_n(x) are related to the Bernoulli numbers by B_n = n!σ_n(1). S. C. Woon (Woon 1997) described an algorithm to compute σ_n(1) as a binary tree.

Woon's tree for σ_n(1)

Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = [1,2]. Given a node N = [a₁,a₂,..., a_k] of the tree, the left child of the node is L(N) = [−a₁,a₂ + 1, a₃, ..., a_k] and the right child R(N) = [a₁,2, a₂, ..., a_k].

Given a node N the factorial of N is defined as

$N! = a_1 \prod_{k=2...\text{length}(N)} a_k! \ .$

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

$B_n = \sum_{N \ \text{node of tree-level}\ n} \frac{n!}{N!} \ .$

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!)).

[edit] Asymptotic approximation

Leonhard Euler expressed the Bernoulli numbers in terms of the Riemann zeta function as

$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left[1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\cdots\;\right].$

It then follows from the Stirling formula that, as n goes to infinity,

$|B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n}.$

[edit] Integral representation and continuation

The integral

$b(s) = 2e^{s i \pi/2}\int_{0}^{\infty} \frac{st^{s}}{1-e^{2\pi t}} \frac{dt}{t}$

has as special values b(2n) = B_2n for n > 0. The integral might be considered as a continuation of the Bernoulli numbers to the complex plane and this was indeed suggested by Peter Luschny in 2004.^{[citation needed]}

For example b(3) = (3/2)ζ(3)Π⁻³Ι and b(5) = −(15/2) ζ(5) Π⁻⁵Ι. Here ζ(n) denotes the Riemann zeta function and Ι the imaginary unit. It is remarkable that already Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated

$p = \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\text{etc.}\ \right) = 0.0581522\ldots \ ,$

$q = \frac{15}{2\pi^{5}}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+\text{etc.}\ \right) = 0.0254132\ldots \ .$

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 > 1.

[edit] 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 E_2n are in magnitude approximately (2/π)(4²ⁿ − 2²ⁿ) times larger than the Bernoulli numbers B_2n. In consequence:

$\pi \ \sim \ 2 \left(2^{2n} - 4^{2n} \right) \frac{B_{2n}}{E_{2n}} \ .$

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 n odd B_n = E_n = 0 (with the exception B₁), it suffices to consider the case when n is even.

$B_{n}=\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{n}{4^n-2^n}E_{k} \quad (n=2,4,6,\ldots)$

$E_{n}=\sum_{k=1}^{n}\binom{n}{k-1}\frac{2^k-4^k}{k} B_{k} \quad (n=2,4,6,\ldots).$

These conversion formulas express an inverse relation 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

$S_n = 2 \left(\frac{2}{\pi}\right)^{n}\sum_{k=-\infty}^{\infty}\left(4k+1\right)^{-n} \quad (k=0,-1,1,-2,2,\ldots)$

and S₁ = 1 by convention (Elkies 2003). 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 (Euler 1735) ‘De summis serierum reciprocarum’ (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few of these numbers are

$S_n = 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots$

The Bernoulli numbers and Euler numbers are best understood as special views of these numbers, selected from the sequence S_n and scaled for use in special applications.

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

$E_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\ \operatorname{even}\right] n! \, S_{n+1} \quad\qquad (n=0,1,\ldots) \ .$

The expression [n even] 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 R_n = 2 S_n / S_n+1 when n is even. The R_n are rational approximations to π and two successive terms always enclose the true value of π. Beginning with n = 1 the sequence starts

$2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385},\ldots \quad \longrightarrow \pi \ .$

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.

[edit] An algorithmic view: the Seidel triangle

The sequence S_n has another unexpected yet important property: The denominators of S_n divide the factorial (n − 1)!. In other words: the numbers T_n = S_n(n − 1)! are integers.

$T_{n} = 1,1,1,2,5,16,61,272,1385,7936,50521,353792,\ldots \quad (n=1,2,\ldots)$

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

$B_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\text{ even}\right] \frac{n }{2^n-4^n}\, T_{n}\ , \quad (n=2,3,\ldots) \ ,$

$E_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\text{ even}\right] T_{n+1} \quad\quad\qquad(n=0,1,\ldots) \ .$

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers E_n are given immediately by T_2n + 1 and the Bernoulli numbers B_2n are obtained from T_2n by some easy shifting, avoiding rational arithmetic.

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

Seidel's algorithm for T_n

[begin] 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 − 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. [end]

Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont (Dumont 1981)) and was rediscovered several times thereafter.

Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz (Knuth & Buckholtz 1967) gave a recurrence equation for the numbers T_2n and recommended this method for computing B_2n and E_2n ‘on electronic computers using only simple operations on integers’.

V. I. Arnold rediscovered Seidel's algorithm in (Arnold 1991) and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.

[edit] 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 (André 1879) & (André 1881). Looking at the first terms of the Taylor expansion of the trigonometric functions tan x and sec x André made a startling discovery.

$\tan x = 1\frac{x}{1!} + 2\frac{x^3}{3!} + 16\frac{x^5}{5!} + 272\frac{x^7}{7!} + 7936\frac{x^9}{9!} + \cdots$

$\sec x = 1 + 1\frac{x^2}{2!} + 5\frac{x^4}{4!} + 61\frac{x^6}{6!} + 1385\frac{x^8}{8!} + 50521\frac{x^{10}}{10!} + \cdots$

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 S_n.

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

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).

[edit] Generalizations by polynomials

The Bernoulli polynomials can be regarded as generalizations of the Bernoulli numbers the same as the Euler polynomials are generalizations of the Euler numbers.

[edit] Arithmetical properties of the Bernoulli numbers

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as B_n = − nζ(1 − n) for integers n ≥ 0 provided for n = 0 and n = 1 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 pB_p−1 is congruent to −1 mod 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.

[edit] The Kummer theorems

The Bernoulli numbers are related to Fermat's last theorem (FLT) by Kummer's theorem (Kummer 1850), which says:

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

Prime numbers with this property are called regular primes (A007703, A000928). Another classical result of Kummer (Kummer 1851) are the following congruences.

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

$\frac{B_{k(p-1)+b}}{k(p-1)+b}\ \equiv \ \frac{B_{b}}{b} \mod p \ .$

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

[edit] p-adic continuity

If b, m and n are positive integers such that m and n are not divisible by p − 1 and $m \equiv n\, \bmod\, p^{b-1}(p-1)$ , then

$(1-p^{m-1}){B_m \over m} \equiv (1-p^{n-1}){B_n \over n} \,\bmod\, p^b.$

Since B_n = —n ζ(1 — n), this can also be written

$(1-p^{-u})\zeta(u) \equiv (1-p^{-v})\zeta(v)\, \bmod \,p^b\,,$

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

[edit] Ramanujan's congruences

The following relations, due to Ramanujan, provide a more efficient method for calculating Bernoulli numbers:

$m\equiv 0\,\bmod\,6\qquad {{m+3}\choose{m}}B_m={{m+3}\over3}-\sum_{j=1}^{m/6}{m+3\choose{m-6j}}B_{m-6j}$

$m\equiv 2\,\bmod\,6\qquad {{m+3}\choose{m}}B_m={{m+3}\over3}-\sum_{j=1}^{(m-2)/6}{m+3\choose{m-6j}}B_{m-6j}$

$m\equiv 4\,\bmod\, 6\qquad{{m+3}\choose{m}}B_m=-{{m+3}\over6}-\sum_{j=1}^{(m-4)/6}{m+3\choose{m-6j}}B_{m-6j}.$

[edit] Von Staudt–Clausen theorem

The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt (von Staudt 1840) and Thomas Clausen (Clausen 1840) independently in 1840. It describes the arithmetical structure of the Bernoulli numbers.

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:

“The denominator of the 2nth Bernoulli number can be found as follows: Add to all divisors of 2n, 1, 2, a, a', ..., 2n the unity, which gives the sequence 2, 3, a + 1, a' + 1, ..., 2n + 1. Select from this sequence only the prime numbers 2, 3, p, p', etc. and build their product.”

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 = 88. It shows that the denominator of B₈₈ is 61410.

Clausen's algorithm for the denominator of B_n
Clausen: function(integer n)	\|	n = 88
S = divisors(n);	\|	{1, 2, 4, 8, 11, 22, 44, 88}
S = map(k → k + 1, S);	\|	{2, 3, 5, 9, 12, 23, 45, 89}
S = select(isprime, S);	\|	{2, 3, 5, 23, 89}
return product(S);	\|	61410

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.

Von Staudt–Clausen representation of B_n
B₀	=	1
B₁	=	1 − 1/2
B₂	=	1 − 1/2 − 1/3
B₄	=	1 − 1/2 − 1/3 − 1/5
B₆	=	1 − 1/2 − 1/3 − 1/7
B₈	=	1 − 1/2 − 1/3 − 1/5
B₁₀	=	1 − 1/2 − 1/3 − 1/11

The theorem affirms the existence of an integer I_n such that

$B_{n}=\frac{I_n}{2} - \sum_{(p-1)|n} \frac1p \quad (n\geq0)\ .$

The sum is over the primes p for which p − 1 divides n. These are the same primes which are employed in the Clausen algorithm. The proposition holds true for all n ≥ 0, not only for even n. I₁ = 2 and for odd n > 1, I_n = 1.

A consequence of the von Staudt–Clausen theorem is: the denominators of the Bernoulli numbers are square-free and for n ≥ 2 divisible by 6.

[edit] Why do the odd Bernoulli numbers vanish?

The sum

$\varphi_k(n) = \sum_{i=0}^n i^k - \frac{n^k}{2}$

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 B_2k+1−m is 0 for m odd and greater than 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 2B_n is an integer. This seems trivial if one knows beforehand that in this case B_n = 0. However, by applying Worpitzky's representation one gets

$2B_n =\sum_{m=0}^n \left(-1\right)^m \frac{2}{m+1}m! \left\{\begin{matrix} n+1 \\ m+1 \end{matrix}\right\} =0\quad\left(n>1\ \text{odd}\right)$

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_n,m be the number of surjective maps from {1, 2, ..., n} to {1, 2, ..., m}, then $S_{n,m}=m! \left\{\begin{matrix} n \\ m \end{matrix}\right\}$ . The last equation can only hold if

$\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).\$

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.

[edit] 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 number. In fact Marcel Riesz (Riesz 1916) proved that the RH is equivalent to the following assertion:

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

Here R(x) is the Riesz function

$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}}. \$

$n^{\overline{k}}$ denotes the rising factorial power in the notation of D. E. Knuth. The number β_n = B_n/n occur frequently in the study of the zeta function and are significant because β_n is a p-integer for primes p where p − 1 does not divide n. The β_n are called divided Bernoulli number.

[edit] History

[edit] 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 Kōwa's Katsuyo Sampo (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 which considered 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, Egypt).

Only during the late sixteenth and early seventeenth centuries mathematicians made 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) played an important role in this development.

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.

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 (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 c-th 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 Kōwa independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.^[1] 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 nowadays 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. To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth (Knuth 1993) a rigorous proof of Faulhaber’s formula was first published by Carl Jacobi in 1834 (Jacobi 1834). Donald E. Knuth's in-depth study of Faulhaber's formula concludes:

“Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B₀, B₁, B₂, … would provide a uniform

$\quad \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1}1B_1n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right)$

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 $\sum n^m$ from polynomials in N to polynomials in n.” (Knuth 1993, p. 14)

[edit] Reconstruction of 'Summae Potestatum'

Jakob Bernoulli's Summae Potestatum, 1713

The Bernoulli numbers were introduced by Jakob Bernoulli in the book Ars Conjectandi 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 = B₂, B = B₄, C = B₆, D = B₈. 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 $c^{\underline{k}}$ . 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 Mathematics Genealogy Project ^[7] shows Leibniz as the doctoral adviser of Jakob Bernoulli. See also the Earliest Uses of Symbols of Calculus^[8].) 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:

$\sum_{0 < k \leq n} k^{c} = \frac{n^{c+1}}{c+1}+\frac{1}{2}n^c+\sum_{k \geq 2}\frac{B_{k}}{k!}c^{\underline{k-1}}n^{c-k+1}\ .$

In fact this formula imperatively suggests to set B₁ = 1/2 when switching from the so-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 $c^{\underline{k-1}}$ has for k = 0 the value 1/(c + 1). (See the explanation in Concrete Mathematics^[9].) Thus Bernoulli's formula can and has to be written

$\sum_{0 < k \leq n} k^{c} = \sum_{k \geq 0}\frac{B_{k}}{k!}c^{\underline{k-1}}n^{c-k+1} \ .$

if B₁ stands for the value Bernoulli himself has given to the coefficient at that position.

[edit] Appendix

[edit] Assorted identities

A compact form of Bernoulli's formula makes use of an abstract symbol B:

$S_m(n) = {1\over{m+1}} [($ B $+ n) m + 1 - B m + 1]$

where the symbol B $k$ that appears during binomial expansion of the parenthesized term is to be replaced by the Bernoulli number $B k$ (and $B_1 = -{1\over 2}$ ). More suggestively and mnemonically, this may be written as a definite integral:

$S_m(n) = \int_0^n ($ B $+x)^m\,dx$

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

$($ 2B $+ 1) m = (2 - 2 m) B m$
Let n be non-negative and even

$\zeta(n) = \frac{\left(-1\right)^{\frac{n}{2}-1}B_n\left(2\pi\right)^n}{2n!}$
The nth cumulant of the uniform probability distribution on the interval [−1, 0] is B_n/n.

Let n¡ = 1/n! and n ≥ 1. Then B_n is n! times the determinant of the following matrix.

B_n =

1
───
n¡

2¡	1¡	0	0	0	...	0
3¡	2¡	1¡	0	0	...	0
4¡	3¡	2¡	1¡	0	...	0
...	...	...	...	...	...	...
(n − 2)¡	...	...	3¡	2¡	1¡	0
(n − 1)¡	(n − 2)¡	...	...	3¡	2¡	1¡
n¡	(n − 1)¡	(n − 2)¡	...	...	3¡	2¡

Thus the determinant is σ_n(1), the Stirling polynomial at x = 1.

Let n ≥ 1.

$\frac{1}{n} \sum_{k=1}^{n}\binom{n}{k}B_{k}B_{n-k}+B_{n-1}=-B_{n} \quad \text{(L. Euler)}$
Let n ≥ 1. Then (von Ettingshausen 1827)

$\sum_{k=0}^{n}\binom{n+1}{k}(n+k+1)B_{n+k}=0$
Let n ≥ 0. Then (Leopold Kronecker 1883)

$B_n = - \sum_{k=1}^{n+1} \frac{(-1)^k}{k} \binom{n+1}{k} \sum_{j=1}^{k} j^n$
Let n ≥ 1 and m ≥ 1. Then (Carlitz 1968)

$(-1)^{m}\sum_{r=0}^m \binom{m}{r}B_{n+r}=(-1)^{n}\sum_{s=0}^{n}\binom{n}{s}B_{m+s}$
Let n ≥ 4 and

$H_{n}=\sum_{1\leq k\leq n}k^{-1}$

the harmonic number. Then

$\frac{n}{2}\sum_{k=2}^{n-2}\frac{B_{n-k}}{n-k}\frac{B_k}{k} - \sum_{k=2}^{n-2} \binom{n}{k}\frac{B_{n-k}}{n-k}B_{k}=H_{n}B_n \qquad\text{(H. Miki, 1978)}$
Let n ≥ 4. Yuri Matiyasevich found (1997)

$(n+2)\sum_{k=2}^{n-2}B_k B_{n-k}-2\sum_{l=2}^{n-2}\binom{n+2}{l} B_l B_{n-l}=n(n+1)B_n$
Let n ≥ 1

$\frac{n}{2}\left(B_{n-1}(x)+\sum_{k=1}^{n-1}\frac{B_{k}(x)}{k} \frac{B_{n-k}(x)}{n-k}\right) -\sum_{k=0}^{n-1}\binom{n}{k}\frac{B_{n-k}} {n-k}B_{k}(x)=H_{n-1}B_{n}(x)$

This is an identity by Faber-Pandharipande-Zagier-Gessel. Choose x = 0 or x = 1 to get a Bernoulli number identity according to your favourite 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.

$\sum_{k=0}^{n}\binom{n}{k} \frac{B_{k}}{n-k+2} = \frac{B_{n+1}}{n+1}$
Let n ≥ 0 and [b] = 1 if b is true, 0 otherwise.

$-1 + \sum_{k=0}^{n}\binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_{k}(1) = 2^{n}$

and

$-1 + \sum_{k=0}^{n}\binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_{k}(0) = [n=0]$

[edit] Values of the Bernoulli numbers

B_n = 0 for all odd n other than 1. B₁ = 1/2 or −1/2 depending on the convention adopted (see above). The first few non-zero Bernoulli numbers (sequences A027641 and A027642 in OEIS) are listed below.

n	Numerator	Denominator	Decimal approximation (N/D)
`0`	`1`	`1`	`+1.00000000000`
`1`	`-1`	`2`	`-0.50000000000`
`2`	`1`	`6`	`+0.16666666667`
`4`	`-1`	`30`	`-0.03333333333`
`6`	`1`	`42`	`+0.02380952381`
`8`	`-1`	`30`	`-0.03333333333`
`10`	`5`	`66`	`+0.07575757576`
`12`	`-691`	`2730`	`-0.25311355311`
`14`	`7`	`6`	`+1.16666666667`
`16`	`−3617`	`510`	`−7.09215686275`
`18`	`43867`	`798`	`+54.9711779448`

[edit] See also

[edit] Notes

^ ^a ^b Selin, H. (1997), p. 891
^ Smith, D. E. (1914), p. 108
^ Note G in the Menabrea reference
^ Concrete Mathematics, (9.67).
^ Concrete Mathematics, (2.44) and (2.52)
^ Concrete Mathematics, Table 254 Euler's triangle.
^ Mathematics Genealogy Project
^ Earliest Uses of Symbols of Calculus
^ Graham, R.; Knuth, D. E.; Patashnik, O. (1989). Concrete Mathematics (2nd ed.). Addison-Wesley. Section 2.51. ISBN 0-201-55802-5.

[edit] References

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[edit] External links

[selin_1997-0] Selin, H. (1997), p. 891

[smith_mikami_1914-1] Smith, D. E. (1914), p. 108

[2] Note G in the Menabrea reference

[3] Concrete Mathematics, (9.67).

[4] Concrete Mathematics, (2.44) and (2.52)

[5] Concrete Mathematics, Table 254 Euler's triangle.

[6] Mathematics Genealogy Project

[7] Earliest Uses of Symbols of Calculus

[8] Graham, R.; Knuth, D. E.; Patashnik, O. (1989). Concrete Mathematics (2nd ed.). Addison-Wesley. Section 2.51. ISBN 0-201-55802-5.

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Contents