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11: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

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§24.15(i) Genocchi Numbers

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§24.15(ii) Tangent Numbers

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§24.15(iii) Stirling Numbers

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§24.15(iv) Fibonacci and Lucas Numbers

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12: 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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13: 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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14: 27.2 Functions

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§27.2(i) Definitions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

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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: 34.10 Zeros

… ►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. Such zeros are called nontrivial zeros. ►For further information, including examples of nontrivial zeros and extensions to $\mathit{9}j$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).

17: 27.13 Functions

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§27.13(i) Introduction

►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

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§27.13(iii) Waring’s Problem

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18: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►

27.9.2 $$(2|p)={(-1)}^{(p^2-1)/8}.$$

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Symbols:

$(n|p)$: Legendre symbol and $p$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …

19: 1.2 Elementary Algebra

… ►The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is … ►The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_1,a_2,\mathrm{\dots },a_n$ are given by … ►If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_1,a_2,\mathrm{\dots },a_n$, and $n$ positive numbers $p_1,p_2,\mathrm{\dots },p_n$ with … ►The dot product notation $𝐮\cdot 𝐯$ is reserved for the physical three-dimensional vectors of (1.6.2). … ►The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. …

20: 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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