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%I #16 Oct 18 2022 13:31:43
%S 1,1,0,1,1,0,1,0,3,0,1,0,2,13,0,1,0,0,6,75,0,1,0,0,6,38,541,0,1,0,0,0,
%T 36,270,4683,0,1,0,0,0,24,150,2342,47293,0,1,0,0,0,0,240,1260,23646,
%U 545835,0,1,0,0,0,0,120,1560,16926,272918,7087261,0,1,0,0,0,0,0,1800,8400,197316,3543630,102247563,0
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j).
%F For k > 0, e.g.f. of column k: 1/(1 - (exp(x) - 1)^k).
%F T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * Stirling2(j,k) * T(n-j,k).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 0, 0, 0, 0, ...
%e 0, 3, 2, 0, 0, 0, ...
%e 0, 13, 6, 6, 0, 0, ...
%e 0, 75, 38, 36, 24, 0, ...
%e 0, 541, 270, 150, 240, 120, ...
%o (PARI) T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2));
%o (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(exp(x+x*O(x^n))-1)^k), n));
%Y Columns k=0-4 give: A000007, A000670, A052841, A353774, A353775.
%Y Cf. A324162, A357293, A357869, A357881.
%K nonn,tabl
%O 0,9
%A _Seiichi Manyama_, Oct 17 2022
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