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A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

%I A293135 #44 Dec 06 2019 08:41:28
%S A293135 1,1,0,1,1,0,1,1,2,0,1,1,3,12,0,1,1,3,12,48,0,1,1,3,13,72,360,0,1,1,3,
%T A293135 13,72,480,2880,0,1,1,3,13,73,500,3780,25200,0,1,1,3,13,73,500,4020,
%U A293135 35280,241920,0,1,1,3,13,73,501,4050,37380,372960,2903040,0
%N A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.
%H A293135 Seiichi Manyama, <a href="/A293135/b293135.txt">Antidiagonals n = 0..139, flattened</a>
%e A293135 Square array begins:
%e A293135    1,   1,   1,   1,   1, ...
%e A293135    0,   1,   1,   1,   1, ...
%e A293135    0,   2,   3,   3,   3, ...
%e A293135    0,  12,  12,  13,  13, ...
%e A293135    0,  48,  72,  72,  73, ...
%e A293135    0, 360, 480, 500, 500, ...
%p A293135 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293135       add(b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A293135     end:
%p A293135 A:= (n, k)-> n!*b(n$2, k):
%p A293135 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Oct 02 2017
%t A293135 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1, k]/j!, {j, 0, Min[k, n/i]}]]];
%t A293135 A[n_, k_] := n! b[n, n, k];
%t A293135 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)
%Y A293135 Columns k=0..5 give A000007, A088311, A293138, A293195, A293196, A293197.
%Y A293135 Rows n=0 gives A000012.
%Y A293135 Main diagonal gives A000262.
%Y A293135 Cf. A293139.
%K A293135 nonn,tabl,look
%O A293135 0,9
%A A293135 _Seiichi Manyama_, Oct 01 2017