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11: 24.15 Related Sequences of Numbers

… ►

24.15.7

B_{n}=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n+1}{k+1}S\left(n+k,k% \right)\bigg{/}\genfrac{(}{)}{0.0pt}{}{n+k}{k},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

►

24.15.8

\sum_{k=0}^{n}(-1)^{n+k}s\left(n+1,k+1\right)B_{k}=\frac{n!}{n+1}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

… ►

24.15.10

\frac{2n-1}{4n}p^{2}B_{2n}\equiv{S\left(p+2n,p-1\right)\pmod{p^{3}}},

2\leq 2n\leq p-3

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\equiv$ : modular equivalence, $n$ : integer and $p$ : prime
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

… ►The Fibonacci numbers are defined by

u_{0}=0

u_{1}=1

, and

u_{n+1}=u_{n}+u_{n-1}

n\geq 1

. The Lucas numbers are defined by

v_{0}=2

v_{1}=1

, and

v_{n+1}=v_{n}+v_{n-1}

n\geq 1

. …

12: Bibliography C

… ►

B. C. Carlson (1979) Computing elliptic integrals by duplication. Numer. Math. 33 (1), pp. 1–16.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview, ZentralBlatt
Referenced by:: §19.19.
Encodings:: BibTeX

… ►

B. C. Carlson (1995) Numerical computation of real or complex elliptic integrals. Numer. Algorithms 10 (1-2), pp. 13–26.

ⓘ

External Links:: ISSN 1017-1398, MathReview, ZentralBlatt
Referenced by:: §19.36(i), §19.36(ii), §19.36(i), §19.36(i), §19.37(iv).
Encodings:: BibTeX

… ►

J. R. Cash and R. V. M. Zahar (1994) A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.

ⓘ

External Links:: ISBN 0-8176-3753-2, MathReview (R. M. M. Mattheij), ZentralBlatt
Referenced by:: §3.6(vii).
Encodings:: BibTeX

… ►

H. H. Chan (1998) On Ramanujan’s cubic transformation formula for

{}_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)

. Math. Proc. Cambridge Philos. Soc. 124 (2), pp. 193–204.

ⓘ

External Links:: ISSN 0305-0041, Document, MathReview (Aleksander Grytczuk), ZentralBlatt
Referenced by:: §15.8(v).
Encodings:: BibTeX

… ►

Cunningham Project (website)

ⓘ

Notes:: Seeks to factor numbers of the form $b^{n}\pm 1$ for $b=2,3,5,6,7,10,11,12$ .
External Links:: Home Page
Referenced by:: 3rd item.
Encodings:: BibTeX

…

13: 26.11 Integer Partitions: Compositions

… ►For example, there are eight compositions of 4:

4,3+1,1+3,2+2,2+1+1,1+2+1,1+1+2

, and

1+1+1+1

. …

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►

F_{1}=1

►

F_{n}=F_{n-1}+F_{n-2}

n\geq 2

… ►

26.11.7

F_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\,\sqrt{5}}.

ⓘ

Symbols:: $n$ : nonnegative integer and $F_{n}$ : Fibonacci numbers
Referenced by:: (18.5.4_5)
Permalink:: http://dlmf.nist.gov/26.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

…

14: About MathML

… ►, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML). … ►

Browsers supporting MathML

►The Firefox browser has traditionally had the strongest support for MathML and its native MathML is used by default. ►Recent enhancements to the WebKit engine now provide support for MathML Core. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. …

15: 24.13 Integrals

… ►For

m,n=1,2,\dotsc

, ►

24.13.6

\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t=\frac{(-1)^{n-% 1}m!n!}{(m+n)!}B_{m+n}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $m$ : integer, $n$ : integer and $t$ : real or complex
A&S Ref:: 23.1.12
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

… ►

24.13.8

\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

… ►For

m,n=1,2,\dotsc

, ►

24.13.11

\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)\,\mathrm{d}t=(-1)^{n}4\frac% {(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $m$ : integer, $n$ : integer and $t$ : real or complex
A&S Ref:: 23.1.12
Referenced by:: §24.13(ii)
Permalink:: http://dlmf.nist.gov/24.13.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

…

16: 26.14 Permutations: Order Notation

… ►The set

\mathfrak{S}_{n}

(§26.13) can be viewed as the collection of all ordered lists of elements of

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

\{\sigma(1)\sigma(2)\cdots\sigma(n)\}

. …Equivalently, this is the sum over

1\leq j<n

of the number of integers less than

\sigma(j)

that lie in positions to the right of the

j

th position:

\mathop{\mathrm{inv}}(35247816)=2+3+1+1+2+2+0=11.

… ►It is also equal to the number of permutations in

\mathfrak{S}_{n}

with exactly

k+1

weak excedances. … ►

26.14.5

\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{x+k}{n}=x% ^{n}.

ⓘ

Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(ii), §26.14 and Ch.26

… ►

26.14.8

\genfrac{<}{>}{0.0pt}{}{n}{k}=(k+1)\genfrac{<}{>}{0.0pt}{}{n-1}{k}+(n-k)% \genfrac{<}{>}{0.0pt}{}{n-1}{k-1},

n\geq 2

ⓘ

Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(iii), §26.14 and Ch.26

…

17: 26.7 Set Partitions: Bell Numbers

… ►

B\left(n\right)

is the number of partitions of

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

. … ►

26.7.1

B\left(0\right)=1,

ⓘ

Symbols:: $B\left(\NVar{n}\right)$ : Bell number
Permalink:: http://dlmf.nist.gov/26.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.7(i), §26.7 and Ch.26

… ►

26.7.4

B\left(n\right)={\mathrm{e}}^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!}=1+\left% \lfloor{\mathrm{e}}^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}\right\rfloor.

ⓘ

Symbols:: $B\left(\NVar{n}\right)$ : Bell number, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.7(i), §26.7 and Ch.26

… ►

26.7.6

B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}B\left(k\right).

ⓘ

Symbols:: $B\left(\NVar{n}\right)$ : Bell number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: Erratum (V1.0.1) for Equation (26.7.6)
Permalink:: http://dlmf.nist.gov/26.7.E6
Encodings:: pMML, png, TeX
Errata (effective with 1.0.1):: Originally appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$ :
$B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}B\left(n\right)$

Reported 2010-11-07 by Layne Watson
See also:: Annotations for §26.7(iii), §26.7 and Ch.26

… ►

26.7.7

B\left(n\right)=\frac{N^{n}{\mathrm{e}}^{N-n-1}}{(1+\ln N)^{1/2}}\left(1+O% \left(\frac{(\ln n)^{1/2}}{n^{1/2}}\right)\right),

n\to\infty

ⓘ

Symbols:: $B\left(\NVar{n}\right)$ : Bell number, $O\left(\NVar{x}\right)$ : order not exceeding, $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $N$
Permalink:: http://dlmf.nist.gov/26.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.7(iv), §26.7 and Ch.26

…

18: 27.14 Unrestricted Partitions

… ►For example,

p\left(5\right)=7

because there are exactly seven partitions of

5

5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1

. … ►with

p\left(0\right)=1

. …where the exponents

1

2

5

7

12

15

\dots

are the pentagonal numbers, defined by … ►Logarithmic differentiation of the generating function

1/\mathit{f}\left(x\right)

leads to another recursion: …where

\sigma_{1}\left(k\right)

is defined by (27.2.10) with

\alpha=1

. …

19: 24.14 Sums

… ►

24.14.4

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

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►

24.14.7

\sum_{j=0}^{m}\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{m}{j}\genfrac{(}{)}{0.0pt}% {}{n}{k}\frac{B_{j}B_{k}}{m+n-j-k+1}=(-1)^{m-1}\frac{m!n!}{(m+n)!}B_{m+n}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $m$ : integer and $n$ : integer
Referenced by:: §24.14(i)
Permalink:: http://dlmf.nist.gov/24.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►Let

\det[a_{r+s}]

denote a Hankel (or persymmetric) determinant, that is, an

(n+1)\times(n+1)

determinant with element

a_{r+s}

in row

r

and column

s

for

r,s=0,1,\dots,n

. … ►

24.14.11

\det[B_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{6}\Bigg{/}\left(% \prod_{k=1}^{2n+1}k!\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

►

24.14.12

\det[E_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{2}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

…

20: 27.18 Methods of Computation: Primes

… ►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of

n-1,n+1

, or both; see Crandall and Pomerance (2005, §§4.1–4.2). These algorithms are used for testing primality of Mersenne numbers,

2^{n}-1

, and Fermat numbers,

2^{2^{n}}+1

. …