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A355609 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k).

%I A355609 #18 Jul 11 2022 03:36:16
%S A355609 1,1,1,1,0,2,1,0,2,6,1,0,0,3,24,1,0,0,6,20,120,1,0,0,0,12,90,720,1,0,
%T A355609 0,0,24,40,594,5040,1,0,0,0,0,60,540,4200,40320,1,0,0,0,0,120,240,
%U A355609 3528,34544,362880,1,0,0,0,0,0,360,1260,25200,316008,3628800,1,0,0,0,0,0,720,1680,28224,263520,3207240,39916800
%N A355609 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k).
%H A355609 Seiichi Manyama, <a href="/A355609/b355609.txt">Antidiagonals n = 0..139, flattened</a>
%F A355609 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.
%F A355609 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(n-k*j)!.
%e A355609 Square array begins:
%e A355609     1,   1,   1,   1,   1,   1, 1, ...
%e A355609     1,   0,   0,   0,   0,   0, 0, ...
%e A355609     2,   2,   0,   0,   0,   0, 0, ...
%e A355609     6,   3,   6,   0,   0,   0, 0, ...
%e A355609    24,  20,  12,  24,   0,   0, 0, ...
%e A355609   120,  90,  40,  60, 120,   0, 0, ...
%e A355609   720, 594, 540, 240, 360, 720, 0, ...
%o A355609 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), abs(stirling(n-k*j, j, 1))/(n-k*j)!);
%Y A355609 Columns k=0..4 give A000142, A066166, A353228, A353229, A354624.
%Y A355609 Cf. A355607, A355610.
%K A355609 nonn,tabl
%O A355609 0,6
%A A355609 _Seiichi Manyama_, Jul 09 2022