# Bernoulli numbers

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##### 1: 24.1 Special Notation
###### BernoulliNumbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
##### 2: 24.14 Sums
###### §24.14 Sums
24.14.10 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!(2m)!}B_{2j}B_{2k}B_{2\ell}B_{2m}=-{2n+3% \choose 3}B_{2n}-\frac{4}{3}n^{2}(2n-1)B_{2n-2}.$
##### 3: 24.19 Methods of Computation
Equations (24.5.3) and (24.5.4) enable $B_{n}$ and $E_{n}$ to be computed by recurrence. …
$B_{2n}=\dfrac{N_{2n}}{D_{2n}}.$
For algorithms for computing $B_{n}$, $E_{n}$, $B_{n}\left(x\right)$, and $E_{n}\left(x\right)$ see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).
###### §24.19(ii) Values of $B_{n}$ Modulo $p$
For number-theoretic applications it is important to compute $B_{2n}\pmod{p}$ for $2n\leq p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0\pmod{p}$. …
##### 4: 24.10 Arithmetic Properties
24.10.1 $B_{2n}+\sum_{(p-1)\mathbin{|}2n}\frac{1}{p}=\hbox{integer},$
The denominator of $B_{2n}$ is the product of all these primes $p$.
##### 5: 24.5 Recurrence Relations
###### §24.5(ii) Other Identities
24.5.6 $\sum_{k=2}^{n}{n\choose k-2}\frac{B_{k}}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$ $n=2,3,\dots$,
24.5.7 $\sum_{k=0}^{n}{n\choose k}\frac{B_{k}}{n+2-k}=\frac{B_{n+1}}{n+1},$ $n=1,2,\dots$,
24.5.8 $\sum_{k=0}^{n}\frac{2^{2k}B_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$ $n=1,2,\dots$.
##### 6: 4.19 Maclaurin Series and Laurent Series
In (4.19.3)–(4.19.9), $B_{n}$ are the Bernoulli numbers and $E_{n}$ are the Euler numbers (§§24.2(i)24.2(ii)).
4.19.3 $\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$,
4.19.4 $\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $0<|z|<\pi$,
4.19.6 $\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,$ $0<|z|<\pi$,
4.19.7 $\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}B_{2% n}}{n(2n)!}z^{2n},$ $|z|<\pi$,
##### 8: 24.15 Related Sequences of Numbers
###### §24.15(ii) Tangent Numbers
24.15.4 $T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},$ $n=1,2,\dots$,
##### 9: 4.33 Maclaurin Series and Laurent Series
4.33.3 $\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$.
For $B_{2n}$ see §24.2(i). …
##### 10: 24.6 Explicit Formulas
###### §24.6 Explicit Formulas
24.6.2 $B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},$