Delta stirlingses 716) 351 6210 Business Class Reservation Booking

SearchAdvancedHelp

Your search matched, but the results seem poor.

The term"bigdelta" was not found.Possible alternative terms: "bigeta", "stirlingsses", "715".

See Search Help for suggestions.

(0.002 seconds)

9 matching pages

1: 26.8 Set Partitions: Stirling Numbers

… ► $s(n,k)$ denotes the Stirling number of the first kind: ${(-1)}^{n-k}$ times the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. … … ►

Table 26.8.1: Stirling numbers of the first kind $s(n,k)$.

► ►►

$n$	$k$
	…

►

… ►Let $A$ and $B$ be the $n\times n$ matrices with $(j,k)$th elements $s(j,k)$, and $S(j,k)$, respectively. … ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

2: 26.1 Special Notation

… ► ►►►

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

… ►Other notations for $s(n,k)$, 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)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). …

3: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ 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, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. …

4: 24.15 Related Sequences of Numbers

… ►The Stirling numbers of the first kind $s(n,m)$, and the second kind $S(n,m)$, are as defined in §26.8(i). … ►

24.15.8 $$\sum _{k=0}^n{(-1)}^{n+k}s(n+1,k+1)B_k=\frac{n!}{n+1}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $s(n,k)$: 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

…

5: 26.13 Permutations: Cycle Notation

… ►

26.13.3 $$\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]=\left|s(n,k)\right|.$$

ⓘ

Defines:

$\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind

Symbols:

$s(n,k)$: Stirling number of the first kind, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§4.13

Permalink:

http://dlmf.nist.gov/26.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

…

6: 24.16 Generalizations

… ►

24.16.8 $${\beta }_n(\lambda )=n!b_n{\lambda }^n+\sum _{k=1}^{\lfloor n/2\rfloor }\frac{n}{2k}{B}_{2k}s(n-1,2k-1){\lambda }^{n-2k},$$ $n=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $k$: integer, $n$: integer and ${\beta }_n(\lambda )$

Referenced by:

§24.16(i)

Permalink:

http://dlmf.nist.gov/24.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

►Here $s(n,m)$ again denotes the Stirling number of the first kind. …

7: Bibliography J

… ►

Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).

ⓘ

Notes:

Translated title: Mathieu functions and Klein-Gordon polynomials.

External Links:

ISSN 0764-4442, Document, MathReview, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

…

8: Bibliography D

… ►

Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.

ⓘ

Notes:

Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997

External Links:

ISBN 0-8218-1198-3, MathReview, ZentralBlatt

Referenced by:

§21.9.

Encodings:

BibTeX

…

9: Errata

… ►

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

$n$	$k$
	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
10	$0$	$-\mathrm{3\hspace{0.17em}62880}$	$\mathrm{10\hspace{0.17em}26576}$	$-\mathrm{11\hspace{0.17em}72700}$	$\mathrm{7\hspace{0.17em}23680}$	$-\mathrm{2\hspace{0.17em}69325}$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 by Svante Janson.

…