A256895 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} j!*C(n-1,j-1)*T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 49, 47, 13, 1, 1, 261, 341, 139, 21, 1, 1, 1631, 2731, 1471, 329, 31, 1, 1, 11743, 24173, 16213, 4789, 671, 43, 1, 1, 95901, 235463, 189373, 69441, 12881, 1231, 57, 1, 1, 876809, 2509621, 2357503, 1032245, 237961, 30169, 2087, 73, 1

Offset: 0

Author

Peter Luschny, Apr 28 2015

Keywords

• nonn
• tabl
• easy

Comments

Can be understood as a convolution matrix or as a sequence-to-triangle transformation similar to the partial Bell polynomials defined as: S -> T(n, k, S) = Sum_{j=0..n-k+1} C(n-1,j-1)*S(j)*T(n-j,k-1,S) if k != 0 else S(0)^n. Here S(n) = n!. The case S(n) = n gives the triangle of idempotent numbers A059297 and the case S(n) = 1 for all n leads to A256894.

Examples

			Triangle starts:
1;
1, 1;
1, 3, 1;
1, 11, 7, 1;
1, 49, 47, 13, 1;
1, 261, 341, 139, 21, 1;
		

Crossrefs

Cf. A001339, A002061, A059297, A256894.

Programs

Maple

# Implemented as a sequence transformation acting on f: n -> n!.
F := proc(n, k, f) option remember; `if`(k=0, f(0)^n,
add(binomial(n-1, j-1)*f(j)*F(n-j, k-1, f), j=0..n-k+1)) end:
for n from 0 to 7 do seq(F(n, k, j->j!), k=0..n) od;

Formula

T(n+1,1) = A001339(n) for n>=0.

T(n,n-1) = A002061(n) for n>=1.