A336187 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^k.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 13, 8, 1, 1, 5, 34, 63, 16, 1, 1, 6, 81, 352, 321, 32, 1, 1, 7, 186, 1685, 3946, 1683, 64, 1, 1, 8, 421, 7416, 38401, 46744, 8989, 128, 1, 1, 9, 946, 30835, 328146, 963525, 573616, 48639, 256, 1, 1, 10, 2113, 122816, 2590225, 16971876, 25346385, 7217536, 265729, 512, 1

Offset: 0

Author

Seiichi Manyama, Jul 11 2020

Keywords

• nonn
• tabl

Comments

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - k * Product_{j=1..k} x_j) for k>0.

Examples

			Square array begins:
  1,  1,    1,     1,      1,        1, ...
  1,  2,    3,     4,      5,        6, ...
  1,  4,   13,    34,     81,      186, ...
  1,  8,   63,   352,   1685,     7416, ...
  1, 16,  321,  3946,  38401,   328146, ...
  1, 32, 1683, 46744, 963525, 16971876, ...
		

Crossrefs

Columns k=0-3 give: A000012, A000079, A001850, A206180.

Main diagonal gives A336188.

Cf. A307883, A336163.

Programs

Mathematica

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[ k^j*Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)