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

1, 1, 0, 1, -1, 1, 1, -3, 5, -2, 1, -6, 17, -20, 9, 1, -10, 45, -100, 109, -44, 1, -15, 100, -355, 694, -689, 265, 1, -21, 196, -1015, 3094, -5453, 5053, -1854, 1, -28, 350, -2492, 10899, -29596, 48082, -42048, 14833

Offset: 1

Author

Thomas Wieder, Dec 29 2006

Keywords

• tabl
• sign

Comments

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.

Examples

			Matrix begins:
1 0 1 -2 9 -44 265 -1854 14833
0 1 -1 5 -20 109 -689 5053 -42048
0 0 1 -3 17 -100 694 -5453 48082
0 0 0 1 -6 45 -355 3094 -29596
0 0 0 0 1 -10 100 -1015 10899
0 0 0 0 0 1 -15 196 -2492
0 0 0 0 0 0 1 -21 350
0 0 0 0 0 0 0 1 -28
0 0 0 0 0 0 0 0 1
		

Crossrefs

Signed version of A094791 [from Olivier Gérard, Jul 31 2011]

Cf. A039810, A039814, A126350, A126351, A054654.

Formula

(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.