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A126353 Triangle read by rows: matrix product of the Stirling numbers of the first kind with the binomial coefficients.

%I A126353 #11 Jan 27 2014 09:13:05
%S A126353 1,1,0,1,-1,1,1,-3,5,-2,1,-6,17,-20,9,1,-10,45,-100,109,-44,1,-15,100,
%T A126353 -355,694,-689,265,1,-21,196,-1015,3094,-5453,5053,-1854,1,-28,350,
%U A126353 -2492,10899,-29596,48082,-42048,14833
%N A126353 Triangle read by rows: matrix product of the Stirling numbers of the first kind with the binomial coefficients.
%C A126353 Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000166 = subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
%F A126353 (In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
%e A126353 Matrix begins:
%e A126353 1 0 1 -2 9 -44 265 -1854 14833
%e A126353 0 1 -1 5 -20 109 -689 5053 -42048
%e A126353 0 0 1 -3 17 -100 694 -5453 48082
%e A126353 0 0 0 1 -6 45 -355 3094 -29596
%e A126353 0 0 0 0 1 -10 100 -1015 10899
%e A126353 0 0 0 0 0 1 -15 196 -2492
%e A126353 0 0 0 0 0 0 1 -21 350
%e A126353 0 0 0 0 0 0 0 1 -28
%e A126353 0 0 0 0 0 0 0 0 1
%Y A126353 Signed version of A094791 [from _Olivier Gérard_, Jul 31 2011]
%Y A126353 Cf. A039810, A039814, A126350, A126351, A054654.
%K A126353 tabl,sign
%O A126353 1,8
%A A126353 _Thomas Wieder_, Dec 29 2006