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A357869 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)/j!.

%I A357869 #18 Jan 05 2024 12:29:43
%S A357869 1,1,0,1,1,0,1,0,2,0,1,0,2,5,0,1,0,0,6,15,0,1,0,0,6,26,52,0,1,0,0,0,
%T A357869 36,150,203,0,1,0,0,0,24,150,962,877,0,1,0,0,0,0,240,900,6846,4140,0,
%U A357869 1,0,0,0,0,120,1560,9366,54266,21147,0,1,0,0,0,0,0,1800,8400,101556,471750,115975,0
%N A357869 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)/j!.
%H A357869 Andrew Howroyd, <a href="/A357869/b357869.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%F A357869 For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).
%F A357869 T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).
%e A357869 Square array begins:
%e A357869   1,  1,   1,   1,   1,   1, ...
%e A357869   0,  1,   0,   0,   0,   0, ...
%e A357869   0,  2,   2,   0,   0,   0, ...
%e A357869   0,  5,   6,   6,   0,   0, ...
%e A357869   0, 15,  26,  36,  24,   0, ...
%e A357869   0, 52, 150, 150, 240, 120, ...
%o A357869 (PARI) T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);
%o A357869 (PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));
%Y A357869 Columns k=0-4 give: A000007, A000110, A052859, A353664, A353665.
%Y A357869 Cf. A324162, A357293, A357868.
%K A357869 nonn,tabl
%O A357869 0,9
%A A357869 _Seiichi Manyama_, Oct 17 2022