binomial coefficients

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1: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►

$m$	$n$
$m$	…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►

$m$	$n$
$m$	…

► … ►

§26.3(iv) Identities

…

2: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

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\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. … ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

3: 24.6 Explicit Formulas

… ►

24.6.1

B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{% 2n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.6.2

B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

… ►

24.6.9

B_{n}=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.6.10

E_{n}=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1)^{j}(2j% +1)^{n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6, §24.6
Permalink:: http://dlmf.nist.gov/24.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.12

E_{2n}=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^% {2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6, §24.6
Permalink:: http://dlmf.nist.gov/24.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

4: 24.5 Recurrence Relations

… ►

24.5.1

\sum_{k=0}^{n-1}{n\choose k}B_{k}\left(x\right)=nx^{n-1},

n=2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

… ►

24.5.3

\sum_{k=0}^{n-1}{n\choose k}B_{k}=0,

n=2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.5.4

\sum_{k=0}^{n}{2n\choose 2k}E_{2k}=0,

n=1,2,\dots

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.5.5

\sum_{k=0}^{n}{n\choose k}2^{k}E_{n-k}+E_{n}=2.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

… ►

24.5.6

\sum_{k=2}^{n}{n\choose k-2}\frac{B_{k}}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},

n=2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

…

5: 24.14 Sums

… ►

24.14.1

\sum_{k=0}^{n}{n\choose k}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),

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.14.2

\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}=(1-n)B_{n}-nB_{n-1}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.14.3

\sum_{k=0}^{n}{n\choose k}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)),

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

… ►

24.14.5

\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

24.14.6

\sum_{k=0}^{n}{n\choose k}2^{k}B_{k}E_{n-k}=2(1-2^{n-1})B_{n}-nE_{n-1}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

…

6: 17.2 Calculus

… ►

§17.2(ii) Binomial Coefficients

►

17.2.27

\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)% _{m}\left(q;q\right)_{n-m}}\\ =\frac{\left(q^{-n};q\right)_{m}(-1)^{m}q^{nm-\genfrac{(}{)}{0.0pt}{}{m}{2}}}{% \left(q;q\right)_{m}},

ⓘ

Defines:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial)
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.28

\lim_{q\to 1}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{(}{)}{0.0pt}{}{n}{m}=% \frac{n!}{m!(n-m)!},

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $q$ : complex base, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(ii), §17.2 and Ch.17

… ►

17.2.30

\genfrac{[}{]}{0.0pt}{}{-n}{m}_{q}=\genfrac{[}{]}{0.0pt}{}{m+n-1}{m}_{q}(-1)^{% m}q^{-mn-\genfrac{(}{)}{0.0pt}{}{m}{2}},

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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

… ►

17.2.36

\sum_{j=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}(-z)^{j}=(1-z)^{n}.

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : nonnegative integer, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.2.E36
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(iii), §17.2 and Ch.17

…

7: 17.3 $q$ -Elementary and $q$ -Special Functions

… ►

17.3.2

E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.

ⓘ

Defines:: $E_{\NVar{q}}\left(\NVar{x}\right)$ : $q$ -exponential function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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

… ►

17.3.7

\beta_{n}\left(x,q\right)=(1-q)^{1-n}\sum_{r=0}^{n}(-1)^{r}\genfrac{(}{)}{0.0% pt}{}{n}{r}\frac{r+1}{(1-q^{r+1})}q^{rx}.

ⓘ

Defines:: $\beta_{\NVar{n}}\left(\NVar{x},\NVar{q}\right)$ : $q$ -Bernoulli polynomial
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

… ►

17.3.8

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

ⓘ

Defines:: $A_{\NVar{m},\NVar{s}}\left(\NVar{q}\right)$ : $q$ -Euler number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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

… ►

17.3.9

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

ⓘ

Defines:: $a_{\NVar{m},\NVar{s}}\left(\NVar{q}\right)$ : $q$ -Stirling number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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

…

8: 12.13 Sums

… ►

12.13.2

U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a+m,x\right),

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.3

V\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(a-m,x\right),

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

… ►

12.13.5

U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right),

\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §12.13(i), §12.13 and Ch.12

…

9: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

26.9.4

\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}},

n\geq 0

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

►is the Gaussian polynomial (or

q

-binomial coefficient); see also §§17.2(i)–17.2(ii). … ►

26.9.5

\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1% +\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m},

ⓘ

Symbols:: $p_{\NVar{k}}\left(\NVar{n}\right)$ : total number of partitions of $n$ into at most $k$ parts, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

►

26.9.6

\sum_{n=0}^{\infty}p_{k}\left(\leq m,n\right)q^{n}=\genfrac{[}{]}{0.0pt}{}{m+k% }{k}_{q}.

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$ : number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

… ►

26.9.7

\sum_{m,n=0}^{\infty}p_{k}\left(\leq m,n\right)x^{k}q^{n}=1+\sum_{k=1}^{\infty% }\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}.

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$ : number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

…

10: 1.2 Elementary Algebra

… ►

§1.2(i) Binomial Coefficients

… ►See also §26.3(i). … ►For complex

z

the binomial coefficient

\genfrac{(}{)}{0.0pt}{}{z}{k}

is defined via (1.2.6). … ►

1.2.3

\genfrac{(}{)}{0.0pt}{}{n}{0}+\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+\genfrac{(}{% )}{0.0pt}{}{n}{n}=2^{n}.

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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:: pMML, png, TeX
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

►

1.2.4

\genfrac{(}{)}{0.0pt}{}{n}{0}-\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+(-1)^{n}% \genfrac{(}{)}{0.0pt}{}{n}{n}=0.

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and $n$ : nonnegative integer
A&S Ref:: 3.1.7
Referenced by:: §1.2(i), §1.2(i), Erratum (V1.0.5) for Subsection 1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

…