%I #18 Jul 11 2022 03:36:16

%S 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 0,0,24,40,594,5040,1,0,0,0,0,60,540,4200,40320,1,0,0,0,0,120,240,

%U 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 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 Seiichi Manyama, <a href="/A355609/b355609.txt">Antidiagonals n = 0..139, flattened</a>

%F 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 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |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 2, 2, 0, 0, 0, 0, 0, ...

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

%e 24, 20, 12, 24, 0, 0, 0, ...

%e 120, 90, 40, 60, 120, 0, 0, ...

%e 720, 594, 540, 240, 360, 720, 0, ...

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

%Y Columns k=0..4 give A000142, A066166, A353228, A353229, A354624.

%Y Cf. A355607, A355610.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Jul 09 2022