A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)*log(1 + x))/(1 - k*log(1 + x)).

1, 1, 1, 1, 1, -1, 1, 1, 1, 2, 1, 1, 3, 2, -6, 1, 1, 5, 14, 4, 24, 1, 1, 7, 38, 86, 14, -120, 1, 1, 9, 74, 384, 664, 38, 720, 1, 1, 11, 122, 1042, 4854, 6136, 216, -5040, 1, 1, 13, 182, 2204, 18344, 73614, 66240, 600, 40320, 1, 1, 15, 254, 4014, 49774, 387512, 1302552, 816672, 6240, -362880

Offset: 0

Author

Seiichi Manyama, Jun 12 2020

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,  1,   1,    1,     1,     1, ...
   1,  1,   1,    1,     1,     1, ...
  -1,  1,   3,    5,     7,     9, ...
   2,  2,  14,   38,    74,   122, ...
  -6,  4,  86,  384,  1042,  2204, ...
  24, 14, 664, 4854, 18344, 49774, ...
		

Crossrefs

Columns k=1..3 give A006252, A308878, A335530.

Main diagonal gives A335529.

Cf. A320080.

Programs

Mathematica

T[0, k_] = 1; T[n_, k_] := Sum[If[k == 0 && j <= 1, 1, k^(j - 1)] * j! * StirlingS1[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

Formula

T(0,k)=1 and T(n,k) = Sum_{j=0..n} j! * k^(j-1) * Stirling1(n,j) for n > 0.