Genocchi numbers

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1: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 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

…

3: Bibliography D

… ►

H. Davenport (2000) Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.

ⓘ

Notes:: Revised and with a preface by Hugh L. Montgomery
External Links:: ISBN 0-387-95097-4, MathReview, ZentralBlatt
Referenced by:: §27.12.
Encodings:: BibTeX

… ►

J. B. Dence and T. P. Dence (1999) Elements of the Theory of Numbers. Harcourt/Academic Press, San Diego, CA.

ⓘ

External Links:: ISBN 0-12-209130-2, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: Ch.24.
Encodings:: BibTeX

… ►

K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

►

K. Dilcher (2002) Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

ⓘ

External Links:: ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

D. Dumont and G. Viennot (1980) A combinatorial interpretation of the Seidel generation of Genocchi numbers. Ann. Discrete Math. 6, pp. 77–87.

ⓘ

Notes:: Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978)
External Links:: MathReview (J. M. Gandhi), ZentralBlatt
Referenced by:: §24.15(i).
Encodings:: BibTeX

…

4: 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).

5: 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

. …

6: 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

…

7: 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

…

8: 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}