A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 9, 16, 5, 1, 2, 17, 82, 65, 6, 1, 2, 33, 460, 1313, 326, 7, 1, 2, 65, 2674, 29441, 32826, 1957, 8, 1, 2, 129, 15796, 684545, 3680126, 1181737, 13700, 9, 1, 2, 257, 94042, 16175105, 427840626, 794907217, 57905114, 109601, 10

Offset: 0

Author

Seiichi Manyama, May 02 2021

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,   1,     1,       1,         1,           1, ...
  2,   2,     2,       2,         2,           2, ...
  3,   5,     9,      17,        33,          65, ...
  4,  16,    82,     460,      2674,       15796, ...
  5,  65,  1313,   29441,    684545,    16175105, ...
  6, 326, 32826, 3680126, 427840626, 50547203126, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..59, flattened

Crossrefs

Columns 0..3 give A000027(n+1), A000522, A006040, A217284.

Main diagonal gives A336247.

Cf. A291556.

Programs

Mathematica

T[n_, k_] := Sum[(n!/j!)^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)

PARI

T(n, k) = sum(j=0, n, (n!/j!)^k);

Formula

T(0,k) = 1 and T(n,k) = n^k * T(n-1,k) + 1 for n > 0.