Search: id:A048994
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%I A048994
%S A048994 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,
%T A048994 1764,1624,735,175,21,1,0,5040,13068,13132,6769,1960,322,28,
%U A048994 1,0,40320,109584,118124,67284,22449,4536,546,36,1,0,362880,1026576
%V A048994 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,
%W A048994 -1764,1624,-735,175,-21,1,0,-5040,13068,-13132,6769,-1960,322,-28,
%X A048994 1,0,40320,-109584,118124,-67284,22449,-4536,546,-36,1,0,-362880,1026576
%N A048994 Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0<=k<=n.
%C A048994 The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| 
               = number of permutations of n objects with exactly k cycles.
%C A048994 Mirror image of the triangle A054654 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 30 2006
%D A048994 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.
%D A048994 L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.
%D A048994 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 
               1996, p. 93.
%D A048994 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 226.
%D A048994 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 245.
%D A048994 J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%H A048994 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A048994 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling1.html">
               Stirling numbers of the first kind</a>
%F A048994 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.
%F A048994 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.
%F A048994 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.
%F A048994 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
%F A048994 A008275*A007318 as infinite lower triangular matrices. [From Gerald McGarvey 
               (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]
%e A048994 1; 0,1; 0,-1,1; 0,2,-3,1; 0,-6,11,-6,1; 0,24,-50,35,-10,1; ...
%p A048994 A048994 := proc(n,k) combinat[stirling1](n,k) ; end: - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Feb 23 2009
%p A048994 seq(print(seq(coeff(expand(k!*binomial(x,k)),x,i),i=0..k)),k=0..9); [From 
               Peter Luschny (peter(AT)luschny.de), Jul 13 2009]
%o A048994 (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)))
%Y A048994 See especially A008275 which is the main entry for this triangle. Cf. 
               A008277, A039814-A039817, A048993.
%Y A048994 Cf. A084938.
%Y A048994 A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0.
%Y A048994 Sequence in context: A081247 A144633 A005210 this_sequence A132393 A121434 
               A137329
%Y A048994 Adjacent sequences: A048991 A048992 A048993 this_sequence A048995 A048996 
               A048997
%K A048994 sign,tabl,nice
%O A048994 0,8
%A A048994 N. J. A. Sloane (njas(AT)research.att.com).
%E A048994 Offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 
               23 2009
%E A048994 Formula corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 10 
               2009

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