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A355610 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/k!).

%I A355610 #18 Jul 11 2022 03:36:20
%S A355610 1,1,1,1,0,2,1,0,2,6,1,0,0,3,24,1,0,0,3,20,120,1,0,0,0,6,90,720,1,0,0,
%T A355610 0,4,20,594,5040,1,0,0,0,0,10,180,4200,40320,1,0,0,0,0,5,40,1134,
%U A355610 34544,362880,1,0,0,0,0,0,15,210,7980,316008,3628800,1,0,0,0,0,0,6,70,1904,71280,3207240,39916800
%N A355610 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/k!).
%H A355610 Seiichi Manyama, <a href="/A355610/b355610.txt">Antidiagonals n = 0..139, flattened</a>
%F A355610 T(0,k) = 1 and T(n,k) = (n-1)!/k! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
%F A355610 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).
%e A355610 Square array begins:
%e A355610     1,   1,   1,  1,  1, 1, 1, ...
%e A355610     1,   0,   0,  0,  0, 0, 0, ...
%e A355610     2,   2,   0,  0,  0, 0, 0, ...
%e A355610     6,   3,   3,  0,  0, 0, 0, ...
%e A355610    24,  20,   6,  4,  0, 0, 0, ...
%e A355610   120,  90,  20, 10,  5, 0, 0, ...
%e A355610   720, 594, 180, 40, 15, 6, 0, ...
%o A355610 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));
%Y A355610 Columns k=0..4 give A000142, A066166, A351492, A351493, A355507.
%Y A355610 Cf. A355609, A355619.
%K A355610 nonn,tabl
%O A355610 0,6
%A A355610 _Seiichi Manyama_, Jul 09 2022