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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
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
0,5
COMMENTS
Inspired by a formula of Mélika Tebni in A048993.
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
EXAMPLE
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;
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;
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jan 27 2023
STATUS
approved