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11: 26.6 Other Lattice Path Numbers

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§26.6(ii) Generating Functions

►For sufficiently small $|x|$ and $|y|$, … ►

26.6.6 $$\sum _{n=0}^{\mathrm{\infty }}D(n,n)x^n=\frac{1}{\sqrt{1-6x+x^2}},$$

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Symbols:

$n$: nonnegative integer, $D(m,n)$: Delannoy number and $x$: small

Permalink:

http://dlmf.nist.gov/26.6.E6

Encodings:

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See also:

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

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26.6.7 $$\sum _{n=0}^{\mathrm{\infty }}M(n)x^n=\frac{1-x-\sqrt{1-2x-3x^2}}{2x^2},$$

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Symbols:

$n$: nonnegative integer, $M(n)$: Motzkin number and $x$: small

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E7

Encodings:

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See also:

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

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26.6.8 $$\sum _{n,k=1}^{\mathrm{\infty }}N(n,k)x^n{y}^k=\frac{1-x-xy-\sqrt{{(1-x-xy)}^2-4x^{2}y}}{2x},$$

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Symbols:

$k$: nonnegative integer, $n$: nonnegative integer, $N(n,k)$: Narayana number, $x$: small and $y$: small

Permalink:

http://dlmf.nist.gov/26.6.E8

Encodings:

pMML, png, TeX

See also:

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

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12: 26.7 Set Partitions: Bell Numbers

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§26.7(ii) Generating Function

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13: 26.3 Lattice Paths: Binomial Coefficients

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§26.3(ii) Generating Functions

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14: 14.7 Integer Degree and Order

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§14.7(iv) Generating Functions

… ►For other generating functions see Magnus et al. (1966, pp. 232–233) and Rainville (1960, pp. 163–165, 168, 170–171, 184). …

15: 34.7 Basic Properties: $\mathit{9}j$ Symbol

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§34.7(v) Generating Functions

►For generating functions for the $\mathit{9}j$ symbol see Biedenharn and van Dam (1965, p. 258, eq. (4.37)). …

16: 27.7 Lambert Series as Generating Functions

§27.7 Lambert Series as Generating Functions

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17: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

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§24.2(ii) Euler Numbers and Polynomials

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${(-1)}^n{E}_{2n}>0$ .

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18: 27.14 Unrestricted Partitions

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§27.14(ii) Generating Functions and Recursions

… ►as a generating function for the function $p\left(n\right)$ defined in §27.14(i): ►

27.14.3 $$\frac{1}{f\left(x\right)}=\sum _{n=0}^{\mathrm{\infty }}p\left(n\right)x^n,$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

A&S Ref:

24.2.1 I.B

Permalink:

http://dlmf.nist.gov/27.14.E3

Encodings:

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See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►Logarithmic differentiation of the generating function $1/f\left(x\right)$ leads to another recursion: … ►For further information on partitions and generating functions see Andrews (1976); also §§17.2–17.14, and §§26.9–26.10. …

19: 26.14 Permutations: Order Notation

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§26.14(ii) Generating Functions

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20: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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§26.4(ii) Generating Function

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