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11: 26.14 Permutations: Order Notation

… ►As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. …The Eulerian number is equal to the number of permutations in ${𝔖}_n$ with exactly $k$ excedances. … ►

§26.14(iii) Identities

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12: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

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§26.7(i) Definitions

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§26.7(ii) Generating Function

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§26.7(iii) Recurrence Relation

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§26.7(iv) Asymptotic Approximation

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13: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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§24.14(ii) Higher-Order Recurrence Relations

►In the following two identities, valid for $n\ge 2$, the sums are taken over all nonnegative integers $j,k,\mathrm{\ell }$ with $j+k+\mathrm{\ell }=n$. … ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

14: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. ►

§24.10(ii) Kummer Congruences

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§24.10(iii) Voronoi’s Congruence

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§24.10(iv) Factors

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15: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

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§26.8(i) Definitions

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§26.8(v) Identities

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§26.8(vi) Relations to Bernoulli Numbers

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16: 24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … ►

§24.19(ii) Values of $B_n$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. …

17: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

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§24.2(ii) Euler Numbers and Polynomials

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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18: 24.20 Tables

§24.20 Tables

►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. In Wagstaff (2002) these results are extended to $n=60(2)152$ and $n=40(2)88$, respectively, with further complete and partial factorizations listed up to $n=300$ and $n=200$, respectively. …

19: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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24.5.4 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{2n}{2k}\right)E_{2k}=0,$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

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24.5.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

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§24.5(ii) Other Identities

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§24.5(iii) Inversion Formulas

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20: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … ►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …