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21: 5 Gamma Function

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22: 16.7 Relations to Other Functions

… ►For $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols see Chapter 34. …

23: 35.11 Tables

… ►Tables of zonal polynomials are given in James (1964) for $|\kappa |\le 6$, Parkhurst and James (1974) for $|\kappa |\le 12$, and Muirhead (1982, p. 238) for $|\kappa |\le 5$. …

24: 26.13 Permutations: Cycle Notation

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26.13.2

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Referenced by:

§26.13, §26.13, §26.13

Permalink:

http://dlmf.nist.gov/26.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

25: 26.2 Basic Definitions

… ►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. …The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►

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

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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6	11	23	1255	40	37338
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26: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$2$	$1$	$6$	$\mathrm{1.66666\hspace{0.33em}6667}$	$\mathrm{\times 10⁻¹}$
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$6$	$1$	$42$	$\mathrm{2.38095\hspace{0.33em}2381}$	$\mathrm{\times 10⁻²}$
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$14$	$7$	$6$	$\mathrm{1.16666\hspace{0.33em}6667}$
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$26$	$\mathrm{85\hspace{0.33em}53103}$	$6$	$\mathrm{1.42551\hspace{0.33em}7167}$	$\mathrm{\times 10⁶}$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$6$	$-61$
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27: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
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4	1	4	6	4	1
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6	1	6	15	20	15	6	1
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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
	0	1	2	3	4	5	6	7	8
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1	1	2	3	4	5	6	7	8	9
2	1	3	6	10	15	21	28	36	45
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28: 6.8 Inequalities

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6.8.3

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

29: Nico M. Temme

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6 Exponential, Logarithmic, Sine, and Cosine Integrals

… ►In November 2015, Temme was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 3, 6, 7, and 12.

30: 3.4 Differentiation

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$B_{-1}^3=-\frac{1}{6}(2-6t+3t^2),$

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$$B_{-3}^6=-\frac{1}{720}(12-8t-45t^2+20t^3+15t^4-6t^5),$$

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$$B_{-1}^6=-\frac{1}{48}(36-72t-39t^2+52t^3+5t^4-6t^5),$$

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$$B_1^6=\frac{1}{48}(36+72t-39t^2-52t^3+5t^4+6t^5),$$

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$$B_3^6=\frac{1}{720}(12+8t-45t^2-20t^3+15t^4+6t^5).$$

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