A351420 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 3, -1, 24, 1, -3, 8, -13, 8, 120, 1, -4, 16, -48, 77, -26, 720, 1, -5, 27, -124, 386, -576, 194, 5040, 1, -6, 41, -259, 1270, -3905, 5219, -1142, 40320, 1, -7, 58, -471, 3244, -16243, 47701, -55567, 9736, 362880

Offset: 1

Author

Seiichi Manyama, Feb 11 2022

Keywords

• sign
• tabl

Examples

			Square array begins:
    1,   1,    1,     1,      1,      1, ...
    1,   0,   -1,    -2,     -3,     -4, ...
    2,   1,    3,     8,     16,     27, ...
    6,  -1,  -13,   -48,   -124,   -259, ...
   24,   8,   77,   386,   1270,   3244, ...
  120, -26, -576, -3905, -16243, -50375, ...
		

Crossrefs

Columns k=1..5 give A000142(n-1), (-1)^(n-1) * A089064(n), A351421, A351422, A351423.

Main diagonal gives A351424.

Cf. A111933, A351429.

Programs

Mathematica

T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Feb 11 2022 *)

PARI

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));

Formula

T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.