%I #15 Jul 13 2022 06:35:26

%S 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 0,0,0,24,40,1524,72792,1,0,0,0,0,60,900,12600,920904,1,0,0,0,0,120,

%U 240,6048,147328,13109088,1,0,0,0,0,0,360,1260,43680,1705536,207360912

%N 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 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 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(n-k*j)!.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 0, 0, 0, 0, 0, 0, ...

%e 3, 2, 0, 0, 0, 0, 0, ...

%e 14, 3, 6, 0, 0, 0, 0, ...

%e 88, 32, 12, 24, 0, 0, 0, ...

%e 694, 150, 40, 60, 120, 0, 0, ...

%e 6578, 1524, 900, 240, 360, 720, 0, ...

%o (PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(n-k*j)!);

%Y Columns k=0..3 give A007840, A052830, A351503, A351504.

%Y Cf. A355609, A355652.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Jul 13 2022