A362834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^j * Stirling1(n-j,j)/(n-j)!.
1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 20, 0, 1, 0, 8, 9, 64, 90, 0, 1, 0, 10, 12, 132, 300, 594, 0, 1, 0, 12, 15, 224, 630, 2568, 4200, 0, 1, 0, 14, 18, 340, 1080, 6642, 20160, 34544, 0, 1, 0, 16, 21, 480, 1650, 13536, 55440, 193856, 316008, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 0, 0, 0, 0, 0, ... 0, 2, 4, 6, 8, 10, ... 0, 3, 6, 9, 12, 15, ... 0, 20, 64, 132, 224, 340, ... 0, 90, 300, 630, 1080, 1650, ...
Crossrefs
Programs
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PARI
T(n, k) = (-1)^n*n!*sum(j=0, n\2, k^j*stirling(n-j, j, 1)/(n-j)!);
Formula
E.g.f. of column k: 1/(1 - x)^(k*x).