A344913 Table read by rows, T(n, k) (for 0 <= k <= n) = (-2)^(n - k)*k!*Stirling2(n, k).

1, 0, 1, 0, -2, 2, 0, 4, -12, 6, 0, -8, 56, -72, 24, 0, 16, -240, 600, -480, 120, 0, -32, 992, -4320, 6240, -3600, 720, 0, 64, -4032, 28896, -67200, 67200, -30240, 5040, 0, -128, 16256, -185472, 653184, -1008000, 766080, -282240, 40320

Offset: 0

Author

Peter Luschny, Aug 14 2021

Keywords

• sign
• tabl

Examples

			Table starts:
[0] 1;
[1] 0,    1;
[2] 0,   -2,     2;
[3] 0,    4,   -12,       6;
[4] 0,   -8,    56,     -72,     24;
[5] 0,   16,  -240,     600,   -480,      120;
[6] 0,  -32,   992,   -4320,   6240,    -3600,    720;
[7] 0,   64, -4032,   28896, -67200,    67200, -30240,    5040;
[8] 0, -128, 16256, -185472, 653184, -1008000, 766080, -282240, 40320.
		

Crossrefs

Cf. A155585 (row sums), A122704 (alternating row sums, signed), A278075 (signed Fubini polynomials), A000142 (main diagonal), A048993 (Stirling2).

Programs

Maple

T := (n, k) -> (-2)^(n - k)*k!*Stirling2(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

PARI

T(n, k) = (-2)^(n - k)*k!*stirling(n, k, 2); \\ Michel Marcus, Aug 14 2021

Formula

T(n, k) = 2^(n - k)*Sum_{j=0..n} (-1)^(n - j)*binomial(k, j)*j^n.

Let row(n, x) be the n-th row polynomial, then B(n) = row(n-1, 1)*n / (4^n - 2^n) is the n-th Bernoulli number (with B(1) = 1/2) for n >= 1.