sums%20of%20products

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11: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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26.4.2 $$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)=\frac{(n_1+n_2+\mathrm{\cdots }+n_k)!}{n_1!n_2!\mathrm{\cdots }{n}_k!}=\prod _{j=1}^{k-1}\left(\genfrac{}{}{0.0pt}{}{n_j+n_{j+1}+\mathrm{\cdots }+n_k}{n_j}\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.2

Referenced by:

§26.4(i)

Permalink:

http://dlmf.nist.gov/26.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(i), §26.4 and Ch.26

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Table 26.4.1: Multinomials and partitions.

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$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
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$5$	$2$	$2^1,3^1$	$10$	$20$	$10$
$5$	$3$	$1^2,3^1$	$20$	$20$	$10$
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26.4.9 $${(x_1+x_2+\mathrm{\cdots }+x_k)}^n=\sum \left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)x_1^{n_1}{x}_2^{n_2}\mathrm{\cdots }{x}_k^{n_k},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.2

Permalink:

http://dlmf.nist.gov/26.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(ii), §26.4 and Ch.26

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26.4.10 $$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_m}{n_1,n_2,\mathrm{\dots },n_m}\right)=\sum _{k=1}^m\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_m-1}{n_1,n_2,\mathrm{\dots },n_{k-1},n_k-1,n_{k+1},\mathrm{\dots },n_m}\right),$$ $n_1,n_2,\mathrm{\dots },n_m\ge 1$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.2

Permalink:

http://dlmf.nist.gov/26.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(iii), §26.4 and Ch.26

12: Bibliography L

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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A note on the computation of integrals involving products of trigonometric and Bessel functions. Math. Comp. 27 (124), pp. 871–872.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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13: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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14: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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Jacobi polynomials. II. An analytic proof of the product formula. SIAM J. Math. Anal. 5, pp. 125–137.

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External Links:

Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.12, §18.17(ii), §18.18(vi).

Encodings:

BibTeX

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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