A363007 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - f^k(x)), where f(x) = exp(x) - 1.
1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 24, 1, 1, 5, 23, 75, 120, 1, 1, 6, 36, 175, 541, 720, 1, 1, 7, 52, 342, 1662, 4683, 5040, 1, 1, 8, 71, 594, 4048, 18937, 47293, 40320, 1, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 362880, 1, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 3628800
Offset: 0
Examples
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 3, 4, 5, 6, 7, ...
6, 13, 23, 36, 52, 71, ...
24, 75, 175, 342, 594, 949, ...
120, 541, 1662, 4048, 8444, 15775, ...
Crossrefs
Programs
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PARI
T(n, k) = if(k==0, n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
Formula
T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.
T(n,k) = A153278(k,n) for n >= 1 and k >= 1.