►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. …

4: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►

Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

►

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

►

Software Associated with Books.

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

…

5: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

… ►

§24.14(ii) Higher-Order Recurrence Relations

►In the following two identities, valid for

n\geq 2

, the sums are taken over all nonnegative integers

j,k,\ell

with

j+k+\ell=n

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

6: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

►

§26.7(i) Definitions

►

B\left(n\right)

is the number of partitions of

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

. … ►

§26.7(ii) Generating Function

… ►For higher approximations to

B\left(n\right)

n\to\infty

see de Bruijn (1961, pp. 104–108).

7: 26.15 Permutations: Matrix Notation

… ►A permutation with restricted position specifies a subset

B\subseteq\{1,2,\ldots,n\}\times\{1,2,\ldots,n\}

. …The number of derangements of

n

is the number of permutations with forbidden positions

B=\{(1,1),(2,2),\ldots,(n,n)\}

. ►Let

r_{j}(B)

be the number of ways of placing

j

nonattacking rooks on the squares of

B

. … ►If

B=B_{1}\cup B_{2}

, where no element of

B_{1}

is in the same row or column as any element of

B_{2}

, then … ►The number of permutations that avoid

B

is …

8: 24.19 Methods of Computation

… ►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. … ►For algorithms for computing

B_{n}

E_{n}

B_{n}\left(x\right)

, and

E_{n}\left(x\right)

see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. … ►

•

Buhler et al. (1992) uses the expansion

24.19.3

\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $n$ : integer and $t$ : real or complex
Referenced by:: §24.19(ii)
Permalink:: http://dlmf.nist.gov/24.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

…

9: DLMF Project News

error generating summary

10: 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

…

zelle support number %2B%2B1%28888%E2%80%92481%E2%80%924477%29

1—10 of 855 matching pages

1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

2: Customize DLMF

3: About MathML

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