A359759 Table read by rows. T(n, k) = (-1)^(n - k) * Sum_{j=k..n} binomial(n, j) * A354794(j, k) * j^(n - j).

1, 0, 1, 0, -3, 1, 0, 13, -9, 1, 0, -103, 79, -18, 1, 0, 1241, -905, 265, -30, 1, 0, -19691, 13771, -4290, 665, -45, 1, 0, 384805, -262885, 82621, -14630, 1400, -63, 1, 0, -8918351, 6007247, -1888362, 353381, -40390, 2618, -84, 1

Offset: 0

Author

Peter Luschny, Jan 27 2023

Keywords

• sign
• tabl

Comments

Inspired by a formula of Mélika Tebni in A048993.

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,         1;
[2] 0,        -3,          1;
[3] 0,        13,         -9,        1;
[4] 0,      -103,         79,      -18,        1;
[5] 0,      1241,       -905,      265,      -30,       1;
[6] 0,    -19691,      13771,    -4290,      665,     -45,      1;
[7] 0,    384805,    -262885,    82621,   -14630,    1400,    -63,    1;
[8] 0,  -8918351,    6007247, -1888362,   353381,  -40390,   2618,  -84,    1;
[9] 0, 238966705, -159432369, 50110705, -9627702, 1206471, -96138, 4494, -108, 1;
		

Crossrefs

Cf. A059297, A354794, A357247, A048993.

Programs

Maple

T := (n, k) -> (-1)^(n - k)*add(binomial(n, j) * A354794(j, k) * j^(n - j), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

Formula

E.g.f. of column k: (exp(LambertW(x*exp(-x))) - 1)^k / k!. (Note that (exp(-LambertW(-x*exp(-x))) - 1)^k / k! is the e.g.f. of column k of Stirling2.) - Mélika Tebni, Jan 27 2023