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A048994
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Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0<=k<=n.
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105
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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
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OFFSET
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0,8
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COMMENTS
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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 DELEHAM, 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
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REFERENCES
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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.
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.
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LINKS
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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
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FORMULA
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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 Deleham'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 DELEHAM, 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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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Table[StirlingS1[n, m], {n, 0, 9}, {m, 0, n}] - Peter Luschny, Dec 30 2010
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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sign,tabl,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Offset corrected by R. J. Mathar, Feb 23 2009
Formula corrected by Philippe DELEHAM, Sep 10 2009
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STATUS
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approved
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