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1: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

… ►For numerical values of $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ and $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ see Tables 26.3.1 and 26.3.2. … ►

§26.3(ii) Generating Functions

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§26.3(iv) Identities

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2: 24.6 Explicit Formulas

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24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.2 $$B_n=\frac{1}{n+1}\sum _{k=1}^n\sum _{j=1}^k{(-1)}^j{j}^n\left(\genfrac{}{}{0.0pt}{}{n+1}{k-j}\right)/\left(\genfrac{}{}{0.0pt}{}{n}{k}\right),$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.3 $$B_{2n}=\sum _{k=1}^n\frac{(k-1)!k!}{(2k+1)!}\sum _{j=1}^k{(-1)}^{j-1}\left(\genfrac{}{}{0pt}{}{2k}{k+j}\right)j^{2n}.$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.4 $$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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3: 24.5 Recurrence Relations

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24.5.1 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)=n{x}^{n-1},$$ $n=2,3,\mathrm{\dots }$,

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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

Permalink:

http://dlmf.nist.gov/24.5.E1

Encodings:

pMML, png, TeX

See also:

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

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24.5.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)+E_n\left(x\right)=2x^n,$$ $n=1,2,\mathrm{\dots }$.

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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

Permalink:

http://dlmf.nist.gov/24.5.E2

Encodings:

pMML, png, TeX

See also:

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

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24.5.3 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k=0,$$ $n=2,3,\mathrm{\dots }$,

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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:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.5.E3

Encodings:

pMML, png, TeX

See also:

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

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24.5.4 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{2n}{2k}\right)E_{2k}=0,$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

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

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.5.E4

Encodings:

pMML, png, TeX

See also:

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

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24.5.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.5.E5

Encodings:

pMML, png, TeX

See also:

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

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4: 1.2 Elementary Algebra

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§1.2(i) Binomial Coefficients

… ►See also §26.3(i). … ►For complex $z$ the binomial coefficient $\left(\genfrac{}{}{0.0pt}{}{z}{k}\right)$ is defined via (1.2.6). ►

Binomial Theorem

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1.2.3 $$\left(\genfrac{}{}{0.0pt}{}{n}{0}\right)+\left(\genfrac{}{}{0.0pt}{}{n}{1}\right)+\mathrm{\cdots }+\left(\genfrac{}{}{0.0pt}{}{n}{n}\right)=2^n.$$

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Symbols:

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

A&S Ref:

3.1.6

Referenced by:

§1.2(i)

Permalink:

http://dlmf.nist.gov/1.2.E3

Encodings:

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Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

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5: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. … ►It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. ►Goldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.

6: 24.14 Sums

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24.14.1 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)B_{n-k}\left(y\right)=n(x+y-1)B_{n-1}\left(x+y\right)-(n-1)B_n\left(x+y\right),$$

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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

Referenced by:

§24.14(i)

Permalink:

http://dlmf.nist.gov/24.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k{B}_{n-k}=(1-n)B_n-n{B}_{n-1}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.3 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(h\right)E_{n-k}\left(x\right)=2(E_{n+1}\left(x+h\right)-(x+h-1)E_n\left(x+h\right)),$$

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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

Permalink:

http://dlmf.nist.gov/24.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(h\right)B_{n-k}\left(x\right)=2^n{B}_n\left(\frac{1}{2}(x+h)\right),$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n\left(x\right)$: 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/24.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.6 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{B}_k{E}_{n-k}=2(1-2^{n-1})B_n-n{E}_{n-1}.$$

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Symbols:

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

Referenced by:

§24.14(i)

Permalink:

http://dlmf.nist.gov/24.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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7: 17.2 Calculus

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§17.2(ii) Binomial Coefficients

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17.2.30 $${\left[\genfrac{}{}{0.0pt}{}{-n}{m}\right]}_q={\left[\genfrac{}{}{0.0pt}{}{m+n-1}{m}\right]}_q{(-1)}^m{q}^{-mn-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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§17.2(iii) Binomial Theorem

… ►In the limit as $q\to 1$, (17.2.35) reduces to the standard binomial theorem … ►When $a=q^{m+1}$, where $m$ is a nonnegative integer, (17.2.37) reduces to the $q$-binomial series …

8: 26.5 Lattice Paths: Catalan Numbers

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

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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

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

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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

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9: 17.3 $q$-Elementary and $q$-Special Functions

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17.3.2 $$E_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{x}^n}{{(q;q)}_n}={(-(1-q)x;q)}_{\mathrm{\infty }}.$$

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Defines:

$E_q\left(x\right)$: $q$-exponential function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.7 $${\beta }_n(x,q)={(1-q)}^{1-n}\sum _{r=0}^n{(-1)}^r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)\frac{r+1}{(1-q^{r+1})}{q}^{rx}.$$

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Defines:

${\beta }_n(x,q)$: $q$-Bernoulli polynomial

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

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17.3.8 $$A_{m,s}\left(q\right)=q^{\left(\genfrac{}{}{0.0pt}{}{s-m}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{m+1}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

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Defines:

$A_{m,s}\left(q\right)$: $q$-Euler number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

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17.3.9 $$a_{m,s}\left(q\right)=\frac{q^{-\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}{(1-q)}^s}{{(q;q)}_s}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{s}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

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Defines:

$a_{m,s}\left(q\right)$: $q$-Stirling number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

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10: 26.17 The Twelvefold Way

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Table 26.17.1: The twelvefold way.

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

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