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)).

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, 0, 0, 0, 4, 20, 1524, 72792, 1, 0, 0, 0, 0, 10, 270, 12600, 920904, 1, 0, 0, 0, 0, 5, 40, 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

Offset: 0

Author

Seiichi Manyama, Jul 13 2022

Keywords

• nonn
• tabl

Examples

			Square array begins:
     1,    1,   1,  1,  1, 1, 1, ...
     1,    0,   0,  0,  0, 0, 0, ...
     3,    2,   0,  0,  0, 0, 0, ...
    14,    3,   3,  0,  0, 0, 0, ...
    88,   32,   6,  4,  0, 0, 0, ...
   694,  150,  20, 10,  5, 0, 0, ...
  6578, 1524, 270, 40, 15, 6, 0, ...
		

Links

• Amiram Eldar, Antidiagonals n = 0..150, flattened

Crossrefs

Columns k=0..3 give A007840, A052830, A351505, A351506.

Cf. A355610, A355665.

Programs

Mathematica

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 *)

PARI

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

Formula

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.

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