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A355665 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k * log(1 - x)).

%I A355665 #15 Jul 13 2022 06:35:26
%S A355665 1,1,1,1,0,3,1,0,2,14,1,0,0,3,88,1,0,0,6,32,694,1,0,0,0,12,150,6578,1,
%T A355665 0,0,0,24,40,1524,72792,1,0,0,0,0,60,900,12600,920904,1,0,0,0,0,120,
%U A355665 240,6048,147328,13109088,1,0,0,0,0,0,360,1260,43680,1705536,207360912
%N A355665 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k * log(1 - x)).
%F A355665 T(0,k) = 1 and T(n,k) = n! * Sum_{j=k+1..n} 1/(j-k) * T(n-j,k)/(n-j)! for n > 0.
%F A355665 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(n-k*j)!.
%e A355665 Square array begins:
%e A355665      1,    1,   1,   1,   1,   1, 1, ...
%e A355665      1,    0,   0,   0,   0,   0, 0, ...
%e A355665      3,    2,   0,   0,   0,   0, 0, ...
%e A355665     14,    3,   6,   0,   0,   0, 0, ...
%e A355665     88,   32,  12,  24,   0,   0, 0, ...
%e A355665    694,  150,  40,  60, 120,   0, 0, ...
%e A355665   6578, 1524, 900, 240, 360, 720, 0, ...
%o A355665 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(n-k*j)!);
%Y A355665 Columns k=0..3 give A007840, A052830, A351503, A351504.
%Y A355665 Cf. A355609, A355652.
%K A355665 nonn,tabl
%O A355665 0,6
%A A355665 _Seiichi Manyama_, Jul 13 2022