A271699 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.
1, 0, 1, 0, 1, 2, 0, 1, 3, 9, 0, 1, 4, 14, 58, 0, 1, 5, 20, 90, 475, 0, 1, 6, 27, 131, 729, 4666, 0, 1, 7, 35, 182, 1064, 7070, 53116, 0, 1, 8, 44, 244, 1494, 10284, 79470, 684762, 0, 1, 9, 54, 318, 2034, 14478, 114918, 1012368, 9833391
Offset: 0
Examples
Triangle starts: 1, 0, 1, 0, 1, 2, 0, 1, 3, 9, 0, 1, 4, 14, 58, 0, 1, 5, 20, 90, 475, 0, 1, 6, 27, 131, 729, 4666, 0, 1, 7, 35, 182, 1064, 7070, 53116
Programs
-
Maple
T := (n,k) -> add(abs(Stirling1(k,j))*binomial(-j,-n)*(-1)^(n-j), j=0..n): seq(seq(T(n,k), k=0..n), n=0..9);
-
Mathematica
Flatten[Table[Sum[(-1)^(n-j)Binomial[-j,-n] Abs[StirlingS1[k,j]],{j,0,n}], {n,0,9},{k,0,n}]]