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A355652 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/k! * log(1 - x)).

%I A355652 #41 Jul 13 2022 12:07:25
%S A355652 1,1,1,1,0,3,1,0,2,14,1,0,0,3,88,1,0,0,3,32,694,1,0,0,0,6,150,6578,1,
%T A355652 0,0,0,4,20,1524,72792,1,0,0,0,0,10,270,12600,920904,1,0,0,0,0,5,40,
%U A355652 1764,147328,13109088,1,0,0,0,0,0,15,210,12600,1705536,207360912,1,0,0,0,0,0,6,70,2464,146880,23681520,3608233056
%N A355652 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/k! * log(1 - x)).
%H A355652 Amiram Eldar, <a href="/A355652/b355652.txt">Antidiagonals n = 0..150, flattened</a>
%F A355652 T(0,k) = 1 and T(n,k) = (n!/k!) * Sum_{j=k+1..n} 1/(j-k) * T(n-j,k)/(n-j)! for n > 0.
%F A355652 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).
%e A355652 Square array begins:
%e A355652      1,    1,   1,  1,  1, 1, 1, ...
%e A355652      1,    0,   0,  0,  0, 0, 0, ...
%e A355652      3,    2,   0,  0,  0, 0, 0, ...
%e A355652     14,    3,   3,  0,  0, 0, 0, ...
%e A355652     88,   32,   6,  4,  0, 0, 0, ...
%e A355652    694,  150,  20, 10,  5, 0, 0, ...
%e A355652   6578, 1524, 270, 40, 15, 6, 0, ...
%t A355652 T[n_, k_] := n! * Sum[j! * Abs[StirlingS1[n - k*j, j]]/(k!^j*(n - k*j)!), {j, 0, Floor[n/(k + 1)]}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 13 2022 *)
%o A355652 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));
%Y A355652 Columns k=0..3 give A007840, A052830, A351505, A351506.
%Y A355652 Cf. A355610, A355665.
%K A355652 nonn,tabl
%O A355652 0,6
%A A355652 _Seiichi Manyama_, Jul 13 2022