Vandermonde%20sum

SearchAdvancedHelp

(0.001 seconds)

11—20 of 422 matching pages

11: 26.3 Lattice Paths: Binomial Coefficients

… ►

26.3.3 $$\sum _{n=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)x^n={(1+x)}^m,$$ $m=0,1,\mathrm{\dots }$,

ⓘ

Symbols:

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

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

►

26.3.4 $$\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)x^m=\frac{1}{{(1-x)}^{n+1}},$$ .

ⓘ

Symbols:

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

A&S Ref:

24.1.1 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

… ►

26.3.7 $$\left(\genfrac{}{}{0.0pt}{}{m+1}{n+1}\right)=\sum _{k=n}^m\left(\genfrac{}{}{0.0pt}{}{k}{n}\right),$$ $m\ge n\ge 0$,

ⓘ

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/26.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iii), §26.3 and Ch.26

►

26.3.8 $$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{m-n-1+k}{k}\right),$$ $m\ge n\ge 0$.

ⓘ

Symbols:

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

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iii), §26.3 and Ch.26

… ►

26.3.10 $$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)=\sum _{k=0}^n{(-1)}^{n-k}\left(\genfrac{}{}{0.0pt}{}{m+1}{k}\right),$$ $m\ge n\ge 0$,

ⓘ

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/26.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iv), §26.3 and Ch.26

…

12: 4.11 Sums

§4.11 Sums

…

13: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …

14: 8 Incomplete Gamma and Related
Functions

…

15: 28 Mathieu Functions and Hill’s Equation

…

16: 26.2 Basic Definitions

… ►A partition of a nonnegative integer $n$ is an unordered collection of positive integers whose sum is $n$. … ►The integers whose sum is $n$ are referred to as the parts in the partition. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

► ►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
3	3	20	627	37	21637
…

►

17: 8.26 Tables

… ►

Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

… ►

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

18: 23 Weierstrass Elliptic and Modular
Functions

…

19: 4.41 Sums

§4.41 Sums

►For sums of hyperbolic functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).

20: 7.15 Sums

§7.15 Sums

►For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).