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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.

%I A363007 #18 May 12 2023 16:27:06
%S A363007 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,
%T A363007 1,1,7,52,342,1662,4683,5040,1,1,8,71,594,4048,18937,47293,40320,1,1,
%U A363007 9,93,949,8444,57437,251729,545835,362880,1,1,10,118,1425,15775,143783,950512,3824282,7087261,3628800
%N 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.
%F A363007 T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.
%F A363007 T(n,k) = A153278(k,n) for n >= 1 and k >= 1.
%e A363007 Square array begins:
%e A363007     1,   1,    1,    1,    1,     1, ...
%e A363007     1,   1,    1,    1,    1,     1, ...
%e A363007     2,   3,    4,    5,    6,     7, ...
%e A363007     6,  13,   23,   36,   52,    71, ...
%e A363007    24,  75,  175,  342,  594,   949, ...
%e A363007   120, 541, 1662, 4048, 8444, 15775, ...
%o A363007 (PARI) T(n, k) = if(k==0, n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
%Y A363007 Columns k=0..5 give A000142, A000670, A083355, A099391, A363008, A363009.
%Y A363007 Main diagonal gives A363010.
%Y A363007 Cf. A153278, A351420, A351429.
%K A363007 nonn,tabl
%O A363007 0,6
%A A363007 _Seiichi Manyama_, May 12 2023