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21: 26.8 Set Partitions: Stirling Numbers

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26.8.20 $$s(n+1,k+1)=n!\sum _{j=k}^n\frac{{(-1)}^{n-j}}{j!}s(j,k),$$

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Symbols:

$s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(iv), §26.8 and Ch.26

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26.8.22 $$S(n,k)=kS(n-1,k)+S(n-1,k-1),$$

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Symbols:

$S(n,k)$: Stirling number of the second kind, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.4

Permalink:

http://dlmf.nist.gov/26.8.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(iv), §26.8 and Ch.26

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26.8.25 $$S(n+1,k+1)=\sum _{j=k}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)S(j,k),$$

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(iv), §26.8 and Ch.26

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26.8.30 $$\sum _{j=k}^{n}s(n+1,j+1)n^{j-k}=s(n,k).$$

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Symbols:

$s(n,k)$: Stirling number of the first kind, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.3

Permalink:

http://dlmf.nist.gov/26.8.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(v), §26.8 and Ch.26

… ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

22: Notices

… ►These software indexes are provided as a service to the user community. NIST expressly does not endorse or recommend any specific product or service. …

23: 24.4 Basic Properties

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24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

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24.4.11 $$\sum _{\begin{array}{c}k=1\\ \left(k,m\right)=1\end{array}}^m{k}^n=\frac{1}{n+1}\sum _{j=1}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{j}\right)\left(\prod _{p|m}(1-p^{n-j})B_{n+1-j}\right)m^j.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\left(m,n\right)$: greatest common divisor (gcd), $j$: integer, $k$: integer, $m$: integer, $n$: integer and $p$: prime

Referenced by:

§24.4(iii)

Permalink:

http://dlmf.nist.gov/24.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

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24.4.26 $$E_n\left(0\right)=-E_n\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n+1},$$ $n>0$.

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.20

Referenced by:

§24.9, Erratum (V1.0.5) for Equation (24.4.26)

Permalink:

http://dlmf.nist.gov/24.4.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

This equation is true only for $n>0$. Previously, $n=0$ was also allowed.

Reported 2012-05-14 by Vladimir Yurovsky

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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24.4.33 $$E_{2n}\left(\frac{1}{6}\right)=E_{2n}\left(\frac{5}{6}\right)=\frac{1+3^{-2n}}{2^{2n+1}}{E}_{2n}.$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

Referenced by:

§24.4(vi)

Permalink:

http://dlmf.nist.gov/24.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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24: 26.14 Permutations: Order Notation

… ►It is also equal to the number of permutations in ${𝔖}_n$ with exactly $k+1$ weak excedances. … ►

26.14.8 $n\ge 2$,

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Symbols:

: Eulerian number, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

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26.14.16 $n\ge 1$.

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Symbols:

: Eulerian number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.14.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iv), §26.14 and Ch.26

25: Charles W. Clark

… ►Civil Service, Archie Mahan Prize of the OSA, the Physical Sciences Award of the Washington Academy of Sciences, the Gold and Silver Medals of the U. …

26: 24.5 Recurrence Relations

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24.5.6 $$\sum _{k=2}^n\left(\genfrac{}{}{0pt}{}{n}{k-2}\right)\frac{B_k}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$$ $n=2,3,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

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24.5.7 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{B_k}{n+2-k}=\frac{B_{n+1}}{n+1},$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

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27: 24.13 Integrals

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24.13.4 $${\int }_0^{1/2}{B}_n\left(t\right)dt=\frac{1-2^{n+1}}{2^n}\frac{B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.5 $${\int }_{1/4}^{3/4}{B}_n\left(t\right)dt=\frac{E_n}{2^{2n+1}}.$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n$: Euler numbers, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.8 $${\int }_0^1{E}_n\left(t\right)dt=-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}{B}_{n+2},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.9 $${\int }_0^{1/2}{E}_{2n}\left(t\right)dt=-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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28: 13.8 Asymptotic Approximations for Large Parameters

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13.8.16 $$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.16), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies,

$$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$

For (13.8.16) combine this differential equation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1).

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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29: 27.14 Unrestricted Partitions

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27.14.4 $$f\left(x\right)=1-x-x^2+x^5+x^7-x^{12}-x^{15}+\mathrm{\cdots }=1+\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\left(x^{\omega (k)}+x^{\omega (-k)}\right),$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $k$: positive integer, $x$: real number and $\omega (k)$: pentagonal numbers

Permalink:

http://dlmf.nist.gov/27.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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27.14.6 $$p\left(n\right)=\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\left(p\left(n-\omega (k)\right)+p\left(n-\omega (-k)\right)\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-5\right)-p\left(n-7\right)+\mathrm{\cdots },$$

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Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer and $\omega (k)$: pentagonal numbers

A&S Ref:

24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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30: 27.3 Multiplicative Properties

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27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

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Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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