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%I #17 Oct 19 2022 11:11:39
%S 1,1,0,1,1,0,1,0,2,0,1,0,2,6,0,1,0,0,6,24,0,1,0,0,6,34,120,0,1,0,0,0,
%T 36,220,720,0,1,0,0,0,24,210,1688,5040,0,1,0,0,0,0,240,1710,14868,
%U 40320,0,1,0,0,0,0,120,2040,17304,147684,362880,0,1,0,0,0,0,0,1800,17640,194712,1631376,3628800,0
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|/j!.
%F For k > 0, e.g.f. of column k: exp((-log(1-x))^k).
%F T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(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, 2, 2, 0, 0, 0, ...
%e 0, 6, 6, 6, 0, 0, ...
%e 0, 24, 34, 36, 24, 0, ...
%e 0, 120, 220, 210, 240, 120, ...
%o (PARI) T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1))/j!);
%o (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(exp((-log(1-x+x*O(x^n)))^k), n));
%Y Columns k=0-5 give: A000007, A000142, (-1)^n * A009199(n), A353344, A353358, A353404.
%Y Cf. A357119, A357869, A357881.
%K nonn,tabl
%O 0,9
%A _Seiichi Manyama_, Oct 18 2022
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