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A344913
Table read by rows, T(n, k) (for 0 <= k <= n) = (-2)^(n - k)*k!*Stirling2(n, k).
0
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
0,5
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.
EXAMPLE
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.
MAPLE
T := (n, k) -> (-2)^(n - k)*k!*Stirling2(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
PROG
(PARI) T(n, k) = (-2)^(n - k)*k!*stirling(n, k, 2); \\ Michel Marcus, Aug 14 2021
CROSSREFS
Cf. A155585 (row sums), A122704 (alternating row sums, signed), A278075 (signed Fubini polynomials), A000142 (main diagonal), A048993 (Stirling2).
Sequence in context: A221609 A160125 A151868 * A052079 A291483 A181295
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 14 2021
STATUS
approved