A369738 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(1 - (1+x)^k).
1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 2, -1, 0, 1, -4, 3, 4, 1, 0, 1, -5, 4, 21, -20, -1, 0, 1, -6, 5, 56, -63, 8, 1, 0, 1, -7, 6, 115, -104, -423, 184, -1, 0, 1, -8, 7, 204, -95, -2464, 1899, -464, 1, 0, 1, -9, 8, 329, 36, -8245, 1696, 15201, -1648, -1, 0
Offset: 0
Examples
Square array T(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 0, -1, -2, -3, -4, -5, -6, ... 0, 1, 2, 3, 4, 5, 6, ... 0, -1, 4, 21, 56, 115, 204, ... 0, 1, -20, -63, -104, -95, 36, ... 0, -1, 8, -423, -2464, -8245, -21096, ... 0, 1, 184, 1899, 1696, -21275, -124344, ...
Crossrefs
Programs
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PARI
a000587(n) = sum(k=0, n, (-1)^k*stirling(n, k, 2)); T(n, k) = sum(j=0, n, k^j*stirling(n, j, 1)*a000587(j));
Formula
T(0,k) = 1; T(n,k) = -k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * T(n-j,k)/(n-j)!.
T(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * A000587(j).