binomial coefficients

SearchAdvancedHelp

(0.003 seconds)

11—20 of 66 matching pages

11: 26.17 The Twelvefold Way

… ►

Table 26.17.1: The twelvefold way.

► ►►►

elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
…
unlabeled	labeled	$\left({\displaystyle \genfrac{}{}{0pt}{}{k+n-1}{n}}\right)$	$\left({\displaystyle \genfrac{}{}{0pt}{}{k}{n}}\right)$	$\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{n-k}}\right)$
…

►

12: 26.5 Lattice Paths: Catalan Numbers

… ►

26.5.1 $$C\left(n\right)=\frac{1}{n+1}\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)=\frac{1}{2n+1}\left(\genfrac{}{}{0.0pt}{}{2n+1}{n}\right)=\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)-\left(\genfrac{}{}{0.0pt}{}{2n}{n-1}\right)=\left(\genfrac{}{}{0.0pt}{}{2n-1}{n}\right)-\left(\genfrac{}{}{0.0pt}{}{2n-1}{n+1}\right).$$

ⓘ

Defines:

$C\left(n\right)$: Catalan number

Symbols:

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

Referenced by:

§26.5(iv)

Permalink:

http://dlmf.nist.gov/26.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(i), §26.5 and Ch.26

… ►

26.5.5 $$C\left(n+1\right)=\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0.0pt}{}{n}{2k}\right)2^{n-2k}C\left(k\right).$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

…

13: 24.4 Basic Properties

… ►

24.4.12 $$B_n\left(x+h\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)h^{n-k},$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.7

Permalink:

http://dlmf.nist.gov/24.4.E12

Encodings:

pMML, png, TeX

See also:

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

►

24.4.13 $$E_n\left(x+h\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)h^{n-k},$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.7

Permalink:

http://dlmf.nist.gov/24.4.E13

Encodings:

pMML, png, TeX

See also:

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

►

24.4.14 $$E_{n-1}\left(x\right)=\frac{2}{n}\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)(1-2^k)B_k{x}^{n-k},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.4.E14

Encodings:

pMML, png, TeX

See also:

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

… ►

24.4.16 $$E_{2n}=\frac{1}{2n+1}-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k-1}-1)B_{2k}}{k},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/24.4.E16

Encodings:

pMML, png, TeX

See also:

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

►

24.4.17 $$E_{2n}=1-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k}-1)B_{2k}}{2k}.$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/24.4.E17

Encodings:

pMML, png, TeX

See also:

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

…

14: 26.6 Other Lattice Path Numbers

… ►

26.6.1 $$D(m,n)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{m+n-k}{n}\right)=\sum _{k=0}^n{2}^k\left(\genfrac{}{}{0.0pt}{}{m}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n}{k}\right).$$

ⓘ

Defines:

$D(m,n)$: Delannoy number (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… ►

26.6.2 $$M(n)=\sum _{k=0}^n\frac{{(-1)}^k}{n+2-k}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{2n+2-2k}{n+1-k}\right).$$

ⓘ

Defines:

$M(n)$: Motzkin number (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… ►

26.6.3 $$N(n,k)=\frac{1}{n}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{n}{k-1}\right).$$

ⓘ

Defines:

$N(n,k)$: Narayana number (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… ►

26.6.13 $$M(n)=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)C\left(n+1-k\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.14 $$C\left(n\right)=\sum _{k=0}^{2n}{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{2n}{k}\right)M(2n-k).$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

15: 25.6 Integer Arguments

… ►

25.6.8 $$\zeta \left(2\right)=3\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $k$: nonnegative integer

Keywords:

infinite series

Source:

van der Poorten (1980, (2), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

►

25.6.9 $$\zeta \left(3\right)=\frac{5}{2}\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k-1}}{k^3\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $k$: nonnegative integer

Keywords:

infinite series

Source:

van der Poorten (1980, (3), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

►

25.6.10 $$\zeta \left(4\right)=\frac{36}{17}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^4\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $k$: nonnegative integer

Keywords:

infinite series

Source:

van der Poorten (1980, (5), p. 274)

Permalink:

http://dlmf.nist.gov/25.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.13 $${(-1)}^k{\zeta }^{(k)}\left(-2n\right)=\frac{2{(-1)}^n}{{(2\pi )}^{2n+1}}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\mathrm{\Im }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n+1\right){\zeta }^{(m-r)}\left(2n+1\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$

Source:

Apostol (1985a, p. 225)

Permalink:

http://dlmf.nist.gov/25.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

►

25.6.14 $${(-1)}^k{\zeta }^{(k)}\left(1-2n\right)=\frac{2{(-1)}^n}{{(2\pi )}^{2n}}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\mathrm{\Re }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n\right){\zeta }^{(m-r)}\left(2n\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$

Source:

Apostol (1985a, p. 225)

Permalink:

http://dlmf.nist.gov/25.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

…

16: 26.14 Permutations: Order Notation

… ►

26.14.5

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

26.14.6 $n\ge 1$,

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E6

Encodings:

pMML, png, TeX

See also:

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

►

26.14.7

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E7

Encodings:

pMML, png, TeX

See also:

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

… ►

26.14.12 $n\ge m$, $n\ge 1$.

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E12

Encodings:

pMML, png, TeX

See also:

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

… ►

26.14.16 $n\ge 1$.

ⓘ

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

17: 15.17 Mathematical Applications

… ►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

18: 24.15 Related Sequences of Numbers

… ►

24.15.7 $$B_n=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n+1}{k+1}\right)S(n+k,k)/\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right),$$

ⓘ

Symbols:

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

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

… ►

24.15.11 $$\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0pt}{}{n}{2k}\right){\left(\frac{5}{9}\right)}^k{B}_{2k}{u}_{n-2k}=\frac{n}{6}{v}_{n-1}+\frac{n}{3^n}{v}_{2n-2},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: integer, $n$: integer, $u_n$: Fibonacci numbers and $v_n$: Lucas numbers

Permalink:

http://dlmf.nist.gov/24.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iv), §24.15 and Ch.24

►

24.15.12 $$\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0pt}{}{n}{2k}\right){\left(\frac{5}{4}\right)}^k{E}_{2k}{v}_{n-2k}=\frac{1}{2^{n-1}}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: integer, $n$: integer and $v_n$: Lucas numbers

Permalink:

http://dlmf.nist.gov/24.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iv), §24.15 and Ch.24

…

19: 26.8 Set Partitions: Stirling Numbers

… ►

26.8.16 $$-s(n,n-1)=S(n,n-1)=\left(\genfrac{}{}{0.0pt}{}{n}{2}\right),$$

ⓘ

Symbols:

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

A&S Ref:

24.1.3

Permalink:

http://dlmf.nist.gov/26.8.E16

Encodings:

pMML, png, TeX

See also:

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

… ►

26.8.19 $$\left(\genfrac{}{}{0.0pt}{}{k}{h}\right)s(n,k)=\sum _{j=k-h}^{n-h}\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)s(n-j,h)s(j,k-h),$$ $n\ge k\ge h$,

ⓘ

Symbols:

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

A&S Ref:

24.1.3

Permalink:

http://dlmf.nist.gov/26.8.E19

Encodings:

pMML, png, TeX

See also:

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

… ►

26.8.23 $$\left(\genfrac{}{}{0.0pt}{}{k}{h}\right)S(n,k)=\sum _{j=k-h}^{n-h}\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)S(n-j,h)S(j,k-h),$$ $n\ge k\ge h$,

ⓘ

Symbols:

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

A&S Ref:

24.1.4

Permalink:

http://dlmf.nist.gov/26.8.E23

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

… ►

26.8.27 $$s(n,n-k)=\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0.0pt}{}{n-1+j}{k+j}\right)\left(\genfrac{}{}{0.0pt}{}{n+k}{k-j}\right)S(k+j,j),$$

ⓘ

Symbols:

$s(n,k)$: Stirling number of the first kind, $S(n,k)$: 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

A&S Ref:

24.1.3

Permalink:

http://dlmf.nist.gov/26.8.E27

Encodings:

pMML, png, TeX

See also:

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

…

20: 26.11 Integer Partitions: Compositions

… ►

26.11.3 $$c_m\left(n\right)=\left(\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right),$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

…