A360176 Triangle read by rows. T(n, k) = Sum_{j=k..n} binomial(n, j) * (-j)^(n - j) * (-1)^(j - k)* A360177(j, k).

1, 0, 1, 0, -5, 1, 0, 37, -15, 1, 0, -393, 223, -30, 1, 0, 5481, -3815, 745, -50, 1, 0, -95053, 76051, -18870, 1865, -75, 1, 0, 1975821, -1749811, 514381, -65730, 3920, -105, 1, 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1

Offset: 0

Author

Peter Luschny, Jan 28 2023

Keywords

• sign
• tabl

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,         1;
[2] 0,        -5,        1;
[3] 0,        37,      -15,         1;
[4] 0,      -393,      223,       -30,       1;
[5] 0,      5481,    -3815,       745,     -50,       1;
[6] 0,    -95053,    76051,    -18870,    1865,     -75,    1;
[7] 0,   1975821, -1749811,    514381,  -65730,    3920, -105,    1;
[8] 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1;
		

Crossrefs

Cf. A360177, A273954 (column 1), A028895 (subdiagonal).

Programs

Maple

T := (n, k) -> add(binomial(n, j) * (-j)^(n - j) * (-1)^(j - k) * A360177(j, k), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative:
egf := k -> (1 - exp(-LambertW(x*exp(-x))))^k / k!:
ser := k -> series(egf(k), x, 22): T := (n, k) -> n!*coeff(ser(k), x, n):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

Formula

E.g.f. of column k: (1 - exp(-LambertW(x*exp(-x))))^k / k!.