24 Bernoulli and Euler Polynomials Properties 24.14 Sums 24.16 Generalizations

§24.15 Related Sequences of Numbers

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Permalink:: http://dlmf.nist.gov/24.15
See also:: Annotations for Ch.24

§24.15(i) Genocchi Numbers
§24.15(ii) Tangent Numbers
§24.15(iii) Stirling Numbers
§24.15(iv) Fibonacci and Lucas Numbers

§24.15(i) Genocchi Numbers

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Keywords:: Bernoulli numbers, Genocchi numbers, relations to
Notes:: See Dumont and Viennot (1980).
Permalink:: http://dlmf.nist.gov/24.15.i
See also:: Annotations for §24.15 and Ch.24

24.15.1 $\frac{2t}{e^{t}+1}=\sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!},$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : integer, : real or complex and $G_{n}$ : Genocchi numbers
Permalink:: http://dlmf.nist.gov/24.15.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §24.15(i), §24.15 and Ch.24

24.15.2 $G_{n}=2(1-2^{n})B_{n}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, : 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

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Keywords:: Bernoulli numbers, relations to, tangent numbers
Notes:: See Graham et al. (1994, Ch. 6) and Knuth and Buckholtz (1967).
Permalink:: http://dlmf.nist.gov/24.15.ii
See also:: Annotations for §24.15 and Ch.24

24.15.3 $\tan t=\sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!},$

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Symbols:: : factorial (as in ), $\tan\NVar{z}$ : tangent function, : integer, : 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 $T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},$ $n=1,2,\dots$ ,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, : 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 $T_{2n}=0,$ $n=0,1,\dots$ .

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Symbols:: : 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

Table 24.15.1: Genocchi and Tangent numbers.

	1	2	3	4	5	6	7	8
$G_{n}$	1	−1	0	1	0	−3	0	17
$T_{n}$	1	0	2	0	16	0	272	0

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Symbols:: : integer, $G_{n}$ : Genocchi numbers and $T_{n}$ : tangent numbers
Keywords:: Genocchi numbers, table, tables, tangent numbers
Referenced by:: §24.15(i)
Permalink:: http://dlmf.nist.gov/24.15.T1
See also:: Annotations for §24.15(ii), §24.15 and Ch.24

§24.15(iii) Stirling Numbers

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Keywords:: Bernoulli numbers, Stirling numbers, Stirling numbers (first and second kinds), relations to, relations to Bernoulli numbers
Notes:: For (24.15.6) and (24.15.8) see Todorov (1984, pp. 310 and 343). For (24.15.6) and (24.15.7) see Gould (1972, pp. 44 and 48).
Referenced by:: §26.8(vi), Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/24.15.iii
See also:: Annotations for §24.15 and Ch.24

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

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, : factorial (as in ), : integer and : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

24.15.7 $B_{n}=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n+1}{k+1}S\left(n+k,k% \right)\bigg/\genfrac{(}{)}{0.0pt}{}{n+k}{k},$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer and : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

24.15.8 $\sum_{k=0}^{n}(-1)^{n+k}s\left(n+1,k+1\right)B_{k}=\frac{n!}{n+1}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, : factorial (as in ), : integer and : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

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

24.15.9 $p\frac{B_{n}}{n}\equiv S\left(p-1+n,p-1\right)\pmod{p^{2}},$ $1\leq n\leq p-2$ ,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\equiv$ : modular equivalence, : integer and : 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 $\frac{2n-1}{4n}p^{2}B_{2n}\equiv{S\left(p+2n,p-1\right)\pmod{p^{3}}},$ $2\leq 2n\leq p-3$ .

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\equiv$ : modular equivalence, : integer and : 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

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Keywords:: Fibonacci numbers, Lucas numbers
Notes:: See Kelisky (1957, pp. 32 and 34).
Permalink:: http://dlmf.nist.gov/24.15.iv
See also:: Annotations for §24.15 and Ch.24

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

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