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21—30 of 859 matching pages

22: 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

23: 24.6 Explicit Formulas

§24.6 Explicit Formulas

►The identities in this section hold for

n=1,2,\dotsc

. … ►

24.6.4

E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►

24.6.5

E_{2n}=\frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2n}\*\sum_{j=0}^{k}{2% n-2j\choose k-j}2^{j},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.12

E_{2n}=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^% {2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6, §24.6
Permalink:: http://dlmf.nist.gov/24.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►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

… ►

§27.13(iii) Waring’s Problem

…

25: 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.

26: 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).

27: 26.13 Permutations: Cycle Notation

… ►The Stirling cycle numbers of the first kind, denoted by

\genfrac{[}{]}{0.0pt}{}{n}{k}

, count the number of permutations of

\{1,2,\ldots,n\}

with exactly

k

cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►The derangement number,

d(n)

, is the number of elements of

\mathfrak{S}_{n}

with no fixed points: … ►A permutation is even or odd according to the parity of the number of transpositions. …

28: 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

29: 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.
…

30: 24.21 Software

… ►

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$

…

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21—30 of 859 matching pages

21: 24.9 Inequalities

§24.9 Inequalities

22: 26.21 Tables

§26.21 Tables

23: 24.6 Explicit Formulas

§24.6 Explicit Formulas

24: 27.13 Functions

§27.13(i) Introduction

§27.13(ii) Goldbach Conjecture

§27.13(iii) Waring’s Problem

25: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

Prime Number Theorem

26: 26.1 Special Notation

27: 26.13 Permutations: Cycle Notation

28: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

29: 27.1 Special Notation

§27.1 Special Notation

30: 24.21 Software

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$

Cash App Phone Number%E2%98%8E%EF%B8%8F%2B1%28888%E2%80%92481%E2%80%924477%29%E2%98%8E%EF%B8%8F %22Number%22

21—30 of 859 matching pages

21: 24.9 Inequalities

§24.9 Inequalities

22: 26.21 Tables

§26.21 Tables

23: 24.6 Explicit Formulas

§24.6 Explicit Formulas

24: 27.13 Functions

§27.13(i) Introduction

§27.13(ii) Goldbach Conjecture

§27.13(iii) Waring’s Problem

25: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

Prime Number Theorem

26: 26.1 Special Notation

27: 26.13 Permutations: Cycle Notation

28: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

29: 27.1 Special Notation

§27.1 Special Notation

30: 24.21 Software

§24.21(ii) B n , B n ⁡ ( x ) , E n , and E n ⁡ ( x )

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$