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Search: id:A048994
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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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31
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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
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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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 (kolotoko(AT)wanadoo.fr), Dec 30 2006
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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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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
R. M. Dickau, Stirling numbers of the first kind
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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 A000035; Triangle(unsigned) = [0, 1, 1, 2, 2, 3, 3, 4, 4, ...] DELTA A000035; 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 (kolotoko(AT)wanadoo.fr), Oct 29 2005
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EXAMPLE
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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; ...
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PROGRAM
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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)))
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CROSSREFS
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See especially A008275 which is the main entry for this triangle. Cf. A008277, A039814-A039817, A048993.
Cf. A084938.
A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0.
Adjacent sequences: A048991 A048992 A048993 this_sequence A048995 A048996 A048997
Sequence in context: A100329 A081247 A005210 this_sequence A132393 A121434 A137329
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KEYWORD
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sign,tabl,nice
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AUTHOR
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njas
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