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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: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

… ►

s\left(n,k\right)

denotes the Stirling number of the first kind:

(-1)^{n-k}

times the number of permutations of

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

with exactly

k

cycles. … … ►Let

A

and

B

be the

n\times n

matrices with

(j,k)

th elements

s\left(j,k\right)

, and

S\left(j,k\right)

, respectively. … ►For asymptotic approximations for

s\left(n+1,k+1\right)

and

S\left(n,k\right)

that apply uniformly for

1\leq k\leq n

n\to\infty

see Temme (1993) and Temme (2015, Chapter 34). …

3: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…
$s\left(n,k\right)$	Stirling numbers of the first kind.
…

… ►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).

4: 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

5: 24.15 Related Sequences of Numbers

… ►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

►The Stirling numbers of the first kind

s\left(n,m\right)

, and the second kind

S\left(n,m\right)

, are as defined in §26.8(i). … ►

§24.15(iv) Fibonacci and Lucas Numbers

…

6: 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. …

7: 24.16 Generalizations

§24.16 Generalizations

… ►

Polynomials and Numbers of Integer Order

… ►

Bernoulli Numbers of the Second Kind

… ►

Degenerate Bernoulli Numbers

… ►Here

s\left(n,m\right)

again denotes the Stirling number of the first kind. …

8: Bibliography M

… ►

Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

ⓘ

Notes:: A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.
External Links:: Magma
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

L. J. Mordell (1917) On the representation of numbers as a sum of

2r

squares. Quarterly Journal of Math. 48, pp. 93–104.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

C. Mortici (2013b) Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Modelling 57 (5-6), pp. 1360–1363.

ⓘ

External Links:: ISSN 0895-7177, Document, FullText, MathReview (Emil C. Popa)
Referenced by:: §5.6(i).
Encodings:: BibTeX

►

L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.

ⓘ

External Links:: Document, MathReview (N. J. Fine), ZentralBlatt
Referenced by:: §26.1, §26.8(vii).
Encodings:: BibTeX

►

L. Moser and M. Wyman (1958b) Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

ⓘ

External Links:: Document, MathReview (A. Sade), ZentralBlatt
Referenced by:: §26.1, §26.8(vii), Notations S.
Encodings:: BibTeX

…

9: Guide to Searching the DLMF

… ►

term:

a textual word, a number, or a math symbol.

►

phrase:

any double-quoted sequence of textual words and numbers.

… ►

proximity operator:

adj, prec/n, and near/n, where n is any positive natural number.

… ► ►►►

`$`	stands for any number of alphanumeric characters
	(the more conventional `*` is reserved for the multiplication operator)
…

…

… ►This release increments the minor version number and contains considerable additions of new material and clarifications. … ►This release increments the minor version number and contains considerable additions of new material and clarifications. These additions were facilitated by an extension of the scheme for reference numbers; with “_” introducing intermediate numbers. These enable insertions of new numbered objects between existing ones without affecting their permanent identifiers and URLs. … ►

Table 26.8.1

Originally the Stirling number $s\left(10,6\right)$ was given incorrectly as 6327. The correct number is 63273.

$n$	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
10	$0$	$-3\,62880$	$10\,26576$	$-11\,72700$	$7\,23680$	$-2\,69325$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 by Svante Janson.

…