A355007 Triangle read by rows. T(n, k) = n^k * |Stirling1(n, k)|.

1, 0, 1, 0, 2, 4, 0, 6, 27, 27, 0, 24, 176, 384, 256, 0, 120, 1250, 4375, 6250, 3125, 0, 720, 9864, 48600, 110160, 116640, 46656, 0, 5040, 86436, 557032, 1764735, 2941225, 2470629, 823543, 0, 40320, 836352, 6723584, 27725824, 64225280, 84410368, 58720256, 16777216

Offset: 0

Author

Peter Luschny, Jun 17 2022

Keywords

• nonn
• tabl

Examples

			Table T(n, k) begins:
[0] 1;
[1] 0,    1;
[2] 0,    2,     4;
[3] 0,    6,    27,     27;
[4] 0,   24,   176,    384,     256;
[5] 0,  120,  1250,   4375,    6250,    3125;
[6] 0,  720,  9864,  48600,  110160,  116640,   46656;
[7] 0, 5040, 86436, 557032, 1764735, 2941225, 2470629, 823543;
		

Crossrefs

A000142 (column 1), A000407 (row sums), A000312 (main diagonal), A355006.

Cf. A133942.

Programs

Maple

seq(seq(n^k*abs(Stirling1(n, k)), k = 0..n), n = 0..9);

Mathematica

T[n_, k_] := If[n == k == 0, 1, n^k * Abs[StirlingS1[n, k]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 17 2022 *)

Formula

Sum_{k=0..n} (-1)^k * T(n,k) = A133942(n). - Alois P. Heinz, Mar 30 2023

Conjecture: T(n,k) = A056856(n,k)*n. - R. J. Mathar, Mar 31 2023