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A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0<=k<=n. 110
1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -6, 11, -6, 1, 0, 24, -50, 35, -10, 1, 0, -120, 274, -225, 85, -15, 1, 0, 720, -1764, 1624, -735, 175, -21, 1, 0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1, 0, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1, 0, -362880, 1026576 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.

Mirror image of the triangle A054654. - Philippe Deléham, Dec 30 2006

Also the triangle gives coefficients T(n,k) of x^k in the expansion of C(x,n) = (a(k)*x^k)/n!. -  Mokhtar Mohamed, Dec 04 2012

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.

L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245.

J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

LINKS

T. D. Noe, Rows n=0..100 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

R. M. Dickau, Stirling numbers of the first kind [Illustrates the unsigned Stirling cycle numbers A132393.]

NIST Digital Library of Mathematical Functions, Stirling Numbers

Wikipedia, Stirling numbers and exponential generating functions

FORMULA

s(n, k)=s(n-1, k-1)-(n-1)*s(n-1, k), n, k >= 1; s(n, 0)=s(0, k)=0; s(0, 0)=1.

The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0)=a(0, k)=0; a(0, 0)=1.

Triangle (signed) = [0, -1, -1, -2, -2, -3, -3, -4, -4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; Triangle(unsigned) = [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.

Sum_{k=0..n} (-m)^(n-k)*T(n, k) = A000142(n), A001147(n), A007559(n), A007696(n), ... for m = 1, 2, 3, 4, ... .- Philippe Deléham, Oct 29 2005

A008275*A007318 as infinite lower triangular matrices. [From Gerald McGarvey, Aug 20 2009]

Bivariate e.g.f. exp(x*log(1+t)). - Peter Luschny, Dec 30 2010

EXAMPLE

Triangle begins:

n\k 0     1       2       3      4      5      6    7    8   9

0   1

1   0     1

2   0    -1       1

3   0     2      -3       1

4   0    -6      11      -6      1

5   0    24     -50      35    -10      1

6   0  -120     274    -225     85    -15      1

7   0   720   -1764    1624   -735    175    -21    1

8   0 -5040   13068  -13132   6769  -1960    322  -28    1

9   0 40320 -109584  118124 -67284  22449  -4536  546  -36   1

...  Wolfdieter Lang, Aug 22 2012

MAPLE

A048994:= proc(n, k) combinat[stirling1](n, k) end: # R. J. Mathar, Feb 23 2009

seq(print(seq(coeff(expand(k!*binomial(x, k)), x, i), i=0..k)), k=0..9); # Peter Luschny, Jul 13 2009

A048994_row := proc(n) local k; seq(coeff(expand(pochhammer(x-n+1, n)), x, k), k=0..n) end: # Peter Luschny, Dec 30 2010

MATHEMATICA

Table[StirlingS1[n, m], {n, 0, 9}, {m, 0, n}] - Peter Luschny, Dec 30 2010

PROG

(PARI) a(n, k) = if(k<0|k>n, 0, if(n==0, 1, (n-1)*a(n-1, k)+a(n-1, k-1)))

(Maxima) create_list(stirling1(n, k), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */

(Haskell)

a048994 n k = a048994_tabl !! n !! k

a048994_row n = a048994_tabl !! n

a048994_tabl = map fst $ iterate (\(row, i) ->

   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 0)

-- Reinhard Zumkeller, Mar 18 2013

CROSSREFS

See especially A008275 which is the main entry for this triangle. A132393 is an unsigned version.

Cf. A008277, A039814-A039817, A048993, A084938.

A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0.

Sequence in context: A172380 A144633 A005210 * A132393 A121434 A137329

Adjacent sequences:  A048991 A048992 A048993 * A048995 A048996 A048997

KEYWORD

sign,tabl,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Offset corrected by R. J. Mathar, Feb 23 2009

Formula corrected by Philippe Deléham, Sep 10 2009

STATUS

approved

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Last modified October 31 12:53 EDT 2014. Contains 248866 sequences.