A362839 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.
1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 16, 0, 1, 0, 8, 27, 80, 65, 0, 1, 0, 10, 48, 216, 560, 336, 0, 1, 0, 12, 75, 448, 2025, 4512, 1897, 0, 1, 0, 14, 108, 800, 5120, 21708, 40768, 11824, 0, 1, 0, 16, 147, 1296, 10625, 67584, 260253, 407808, 80145, 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, 12, 27, 48, 75, ... 0, 16, 80, 216, 448, 800, ... 0, 65, 560, 2025, 5120, 10625, ...
Crossrefs
Programs
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PARI
T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);
Formula
E.g.f. of column k: exp(x * (exp(k * x) - 1)).
G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).
T(n,k) = Sum_{j=0..n} (k*j-1)^(n-j) * binomial(n,j).