%I #25 Jul 10 2022 08:07:41
%S 1,1,1,1,0,2,1,0,2,5,1,0,0,3,15,1,0,0,6,16,52,1,0,0,0,12,65,203,1,0,0,
%T 0,24,20,336,877,1,0,0,0,0,60,390,1897,4140,1,0,0,0,0,120,120,2562,
%U 11824,21147,1,0,0,0,0,0,360,210,11816,80145,115975,1,0,0,0,0,0,720,840,20496,105912,586000,678570
%N Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).
%H Seiichi Manyama, <a href="/A292892/b292892.txt">Antidiagonals n = 0..139, flattened</a>
%F From _Seiichi Manyama_, Jul 09 2022: (Start)
%F T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
%F T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)
%e Square array begins:
%e 1, 1, 1, 1, 1, ...
%e 1, 0, 0, 0, 0, ...
%e 2, 2, 0, 0, 0, ...
%e 5, 3, 6, 0, 0, ...
%e 15, 16, 12, 24, 0, ...
%e 52, 65, 20, 60, 120, ...
%o (PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ _Seiichi Manyama_, Jul 09 2022
%Y Columns k=0..3 give A000110, A052506, A240989, A292891.
%Y Rows n=0..1 give A000012, A000007.
%Y Main diagonal gives A000007.
%Y Cf. A292894, A355607.
%K nonn,tabl
%O 0,6
%A _Seiichi Manyama_, Sep 26 2017