A350261 Triangle read by rows. T(n, k) = k^n * BellPolynomial(n, -1/k) for k > 0, if k = 0 then T(n, k) = k^n.

1, 0, -1, 0, 0, -1, 0, 1, 1, -1, 0, 1, 9, 19, 25, 0, -2, 23, 128, 343, 674, 0, -9, -25, 379, 2133, 6551, 15211, 0, -9, -583, -1549, 3603, 33479, 123821, 331827, 0, 50, -3087, -32600, -112975, -174114, 120865, 1619108, 5987745

Offset: 0

Author

Peter Luschny, Dec 22 2021

Keywords

• sign
• tabl

Examples

			Triangle starts:
[0] 1
[1] 0, -1
[2] 0,  0,    -1
[3] 0,  1,     1,     -1
[4] 0,  1,     9,     19,      25
[5] 0, -2,    23,    128,     343,     674
[6] 0, -9,   -25,    379,    2133,    6551,  15211
[7] 0, -9,  -583,  -1549,    3603,   33479, 123821,  331827
[8] 0, 50, -3087, -32600, -112975, -174114, 120865, 1619108, 5987745
		

Crossrefs

Cf. A350256, A350257, A350258, A350259, A350260, A350262, A350263.

Cf. A000587, A318183.

Programs

Maple

A350261 := (n, k) -> ifelse(k = 0, k^n, k^n * BellB(n, -1/k)):
seq(seq(A350261(n, k), k = 0..n), n = 0..8);

Mathematica

T[n_, k_] := If[k == 0, k^n, k^n BellB[n, -1/k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten