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11: 26.11 Integer Partitions: Compositions

… ►

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

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

is the number of compositions of

n

with no 1’s, where again

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

. … ►

26.11.1

c\left(0\right)=c\left(\in\!T,0\right)=1.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ and $T$ : set
Permalink:: http://dlmf.nist.gov/26.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

12: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer

x

is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►These algorithms are used for testing primality of Mersenne numbers,

2^{n}-1

, and Fermat numbers,

2^{2^{n}}+1

. …

13: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

… ►

Motzkin Number $M(n)$

… ►

Narayana Number $N(n,k)$

… ►

§26.6(iv) Identities

…

14: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

… ►

§24.15(iv) Fibonacci and Lucas Numbers

…

15: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

►

§26.5(i) Definitions

►

C\left(n\right)

is the Catalan number. … ►

§26.5(ii) Generating Function

… ►

§26.5(iii) Recurrence Relations

…

16: Bibliography W

… ►

E. Wagner (1990) Asymptotische Entwicklungen der Gaußschen hypergeometrischen Funktion für unbeschränkte Parameter. Z. Anal. Anwendungen 9 (4), pp. 351–360 (German).

ⓘ

External Links:: ISSN 0232-2064, MathReview (T. Erber), ZentralBlatt
Referenced by:: §15.12(ii).
Encodings:: BibTeX

… ►

S. S. Wagstaff (2002) Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

ⓘ

External Links:: FullText, MathReview (Arnold M. Adelberg), ZentralBlatt
Referenced by:: §24.20.
Encodings:: BibTeX

… ►

G. N. Watson (1935a) Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.

ⓘ

External Links:: ISSN 0010-437X, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §20.7(v), §20.8.
Encodings:: BibTeX

… ►

R. Wong (1982) Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt
Referenced by:: §10.74(vii), §3.5(vii).
Encodings:: BibTeX

… ►

World Combinatorics Exchange (website)

ⓘ

Notes:: A service of the Electronic Journal of Combinatorics with links to databases, software, and other resources.
External Links:: World Combinatorics Exchange
Referenced by:: 8th item.
Encodings:: BibTeX

…

17: Bibliography F

… ►

H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp(i\theta)$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.
External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii), §19.37(ii).
Encodings:: BibTeX

… ►

S. Fillebrown (1992) Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.

ⓘ

External Links:: ISSN 0196-6774, MathReview (P. K. Subramanian), ZentralBlatt
Referenced by:: §24.19(i).
Encodings:: BibTeX

… ►

A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie (1962) An Index of Mathematical Tables. Vols. I, II. 2nd edition, Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, MA.

ⓘ

Notes:: Table errata: Math. Comp. v.27 (1973), p. 1009.
External Links:: MathReview
Referenced by:: §10.75(i), §11.14(i), §12.19, §13.30, §14.33, §18.41(iii), §19.37(i), §20.15, §22.21, §23.23, §24.20, 4th item, §28.35(iv), §30.17, §33.24, §4.23(viii), §4.46, §5.22(i), §6.19(i), §7.23(i), §8.26(i), §9.18(i).
Encodings:: BibTeX

… ►

A. S. Fokas, B. Grammaticos, and A. Ramani (1993) From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §32.7(ii).
Encodings:: BibTeX

… ►

Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §18.38(ii).
Encodings:: BibTeX

18: 4.19 Maclaurin Series and Laurent Series

… ►In (4.19.3)–(4.19.9),

B_{n}

are the Bernoulli numbers and

E_{n}

are the Euler numbers (§§24.2(i)–24.2(ii)). ►

4.19.3

\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

|z|<\frac{1}{2}\pi

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tan\NVar{z}$ : tangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.67
Referenced by:: §4.19
Permalink:: http://dlmf.nist.gov/4.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.4

\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

0<|z|<\pi

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.68
Referenced by:: §4.33
Permalink:: http://dlmf.nist.gov/4.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.5

\sec z=1+\frac{z^{2}}{2}+\frac{5}{24}z^{4}+\frac{61}{720}z^{6}+\cdots+\frac{(-% 1)^{n}E_{2n}}{(2n)!}z^{2n}+\cdots,

|z|<\frac{1}{2}\pi

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\sec\NVar{z}$ : secant function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.69
Referenced by:: §24.1, §24.19(i), §24.2(ii)
Permalink:: http://dlmf.nist.gov/4.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.6

\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,

0<|z|<\pi

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.70
Permalink:: http://dlmf.nist.gov/4.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

…

19: 26.14 Permutations: Order Notation

… ►As an example,

35247816

is an element of

\mathfrak{S}_{8}.

The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted

\genfrac{<}{>}{0.0pt}{}{n}{k}

, is the number of permutations in

\mathfrak{S}_{n}

with exactly

k

descents. …The Eulerian number

\genfrac{<}{>}{0.0pt}{}{n}{k}

is equal to the number of permutations in

\mathfrak{S}_{n}