A335095 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n+1)!!)^k * Sum_{j=1..n} 1/(2*j+1)^k.

0, 0, 1, 0, 1, 2, 0, 1, 8, 3, 0, 1, 34, 71, 4, 0, 1, 152, 1891, 744, 5, 0, 1, 706, 55511, 164196, 9129, 6, 0, 1, 3368, 1745731, 41625144, 20760741, 129072, 7, 0, 1, 16354, 57365351, 11575291716, 56246975289, 3616621254, 2071215, 8

Offset: 0

Author

Seiichi Manyama, Sep 12 2020

Keywords

• nonn
• tabl

Examples

			Square array begins:
  0,   0,      0,        0,           0, ...
  1,   1,      1,        1,           1, ...
  2,   8,     34,      152,         706, ...
  3,  71,   1891,    55511,     1745731, ...
  4, 744, 164196, 41625144, 11575291716, ...
		

Crossrefs

Column k=0..4 give A001477, A334670, A335090, A335091, A335092.

Rows n=0-2 give: A000004, A000012, A074606.

Main diagonal gives A335096.

Cf. A291656.

Programs

Mathematica

T[n_, k_] := ((2*n + 1)!!)^k * Sum[1/(2*j + 1)^k, {j, 1, n}]; Table[T[k, n - k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

PARI

{T(n, k) = prod(j=1, n, 2*j+1)^k*sum(j=1, n, 1/(2*j+1)^k)}

Formula

T(0,k) = 0, T(1,k) = 1 and T(n,k) = ((2*n-1)^k+(2*n+1)^k) * T(n-1,k) - (2*n-1)^(2*k) * T(n-2, k) for n>1.