# §24.15 Related Sequences of Numbers

## §24.15(i) Genocchi Numbers

 24.15.1 $\displaystyle\frac{2t}{e^{t}+1}$ $\displaystyle=\sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!},$ 24.15.2 $\displaystyle G_{n}$ $\displaystyle=2(1-2^{n})B_{n}.$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $n$: integer and $G_{n}$: Genocchi numbers Permalink: http://dlmf.nist.gov/24.15.E2 Encodings: TeX, pMML, png See also: Annotations for §24.15(i), §24.15 and Ch.24

See Table 24.15.1.

## §24.15(ii) Tangent Numbers

 24.15.3 $\tan t=\sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!},$ ⓘ Symbols: $!$: factorial (as in $n!$), $\tan\NVar{z}$: tangent function, $n$: integer, $t$: real or complex and $T_{n}$: tangent numbers Referenced by: §24.19(i) Permalink: http://dlmf.nist.gov/24.15.E3 Encodings: TeX, pMML, png See also: Annotations for §24.15(ii), §24.15 and Ch.24
 24.15.4 $T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},$ $n=1,2,\dots$, ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $n$: integer and $T_{n}$: tangent numbers Referenced by: §24.19(i) Permalink: http://dlmf.nist.gov/24.15.E4 Encodings: TeX, pMML, png See also: Annotations for §24.15(ii), §24.15 and Ch.24
 24.15.5 $T_{2n}=0,$ $n=0,1,\dots$. ⓘ Symbols: $n$: integer and $T_{n}$: tangent numbers Permalink: http://dlmf.nist.gov/24.15.E5 Encodings: TeX, pMML, png See also: Annotations for §24.15(ii), §24.15 and Ch.24

## §24.15(iii) Stirling Numbers

The Stirling numbers of the first kind $s\left(n,m\right)$, and the second kind $S\left(n,m\right)$, are as defined in §26.8(i).

 24.15.6 $\displaystyle B_{n}$ $\displaystyle=\sum_{k=0}^{n}(-1)^{k}\frac{k!S\left(n,k\right)}{k+1},$ 24.15.7 $\displaystyle B_{n}$ $\displaystyle=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n+1}{k+1}S\left(n+% k,k\right)\bigg{/}\genfrac{(}{)}{0.0pt}{}{n+k}{k},$ 24.15.8 $\displaystyle\sum_{k=0}^{n}(-1)^{n+k}s\left(n+1,k+1\right)B_{k}=\frac{n!}{n+1}.$

In (24.15.9) and (24.15.10) $p$ denotes a prime. See Horata (1991).

 24.15.9 $p\frac{B_{n}}{n}\equiv S\left(p-1+n,p-1\right)\pmod{p^{2}},$ $1\leq n\leq p-2$, ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E9 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24
 24.15.10 $\frac{2n-1}{4n}p^{2}B_{2n}\equiv{S\left(p+2n,p-1\right)\pmod{p^{3}}},$ $2\leq 2n\leq p-3$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E10 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24

## §24.15(iv) Fibonacci and Lucas Numbers

The Fibonacci numbers are defined by $u_{0}=0$, $u_{1}=1$, and $u_{n+1}=u_{n}+u_{n-1}$, $n\geq 1$. The Lucas numbers are defined by $v_{0}=2$, $v_{1}=1$, and $v_{n+1}=v_{n}+v_{n-1}$, $n\geq 1$.

 24.15.11 $\displaystyle\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}% \left(\frac{5}{9}\right)^{k}B_{2k}u_{n-2k}$ $\displaystyle=\frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2},$ 24.15.12 $\displaystyle\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}% \left(\frac{5}{4}\right)^{k}E_{2k}v_{n-2k}$ $\displaystyle=\frac{1}{2^{n-1}}.$

For further information on the Fibonacci numbers see §26.11.