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21: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

22: 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

… ►

§24.10(iii) Voronoi’s Congruence

… ►

§24.10(iv) Factors

…

23: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

… ►

24.14.2

\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}=(1-n)B_{n}-nB_{n-1}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►

§24.14(ii) Higher-Order Recurrence Relations

… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

24: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$\genfrac{<}{>}{0.0pt}{}{m}{n}$	Eulerian number.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…

… ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

25: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for

\nu\left(n\right)

\Lambda\left(n\right)

p_{n}

, and

\pi\left(x\right)

, the functions in §27.2 are multiplicative, which means

f(1)=1

and … ►

27.3.2

f(n)=\prod_{r=1}^{\nu\left(n\right)}f(p^{a_{r}}_{r}).

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20, §27.3
Permalink:: http://dlmf.nist.gov/27.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.6

\sigma_{\alpha}\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}\frac{p^{\alpha(1% +a_{r})}_{r}-1}{p^{\alpha}_{r}-1},

\alpha\neq 0

ⓘ

Symbols:: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers and $a_{r}$ : positive exponent
A&S Ref:: 24.3.3 I.C
Permalink:: http://dlmf.nist.gov/27.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.10

f(n)=\prod_{r=1}^{\nu\left(n\right)}\left(f(p_{r})\right)^{a_{r}}.

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20
Permalink:: http://dlmf.nist.gov/27.3.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

26: 27.1 Special Notation

§27.1 Special Notation

… ► ►►►►

$d, k, m, n$	positive integers (unless otherwise indicated).
…
$p,p_{1},p_{2},\dots$	prime numbers (or primes): integers ( $>1$ ) with only two positive integer divisors, $1$ and the number itself.
…
$x, y$	real numbers.
…

27: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

… ►

Prime Number Theorem

… ►The number of such primes not exceeding

x

is … ►There are infinitely many Carmichael numbers.

28: Bibliography P

… ►

PARI-GP (free interactive system and C library)

ⓘ

Notes:: Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

R. Piessens and M. Branders (1985) A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

ⓘ

Notes:: International conference on special functions: theory and computation (Turin, 1984)
External Links:: ISSN 0373-1243, MathReview, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

… ►

S. Porubský (1998) Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.10(iii).
Encodings:: BibTeX

…

29: 27.2 Functions

… ►

§27.2(i) Definitions

… ►where

p_{1},p_{2},\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

…

30: 24.5 Recurrence Relations

§24.5 Recurrence Relations

… ►

24.5.3

\sum_{k=0}^{n-1}{n\choose k}B_{k}=0,

n=2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(i), §24.5 and Ch.24

… ►

24.5.5

\sum_{k=0}^{n}{n\choose k}2^{k}E_{n-k}+E_{n}=2.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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

►

§24.5(ii) Other Identities

… ►

§24.5(iii) Inversion Formulas

…