A336203 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j)^k.

1, 1, 3, 1, 3, 7, 1, 3, 9, 15, 1, 3, 13, 27, 31, 1, 3, 21, 63, 81, 63, 1, 3, 37, 171, 321, 243, 127, 1, 3, 69, 495, 1521, 1683, 729, 255, 1, 3, 133, 1467, 7761, 14283, 8989, 2187, 511, 1, 3, 261, 4383, 41361, 131283, 138909, 48639, 6561, 1023, 1, 3, 517, 13131, 227601, 1256283, 2336629, 1385163, 265729, 19683, 2047

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) - 2 * Product_{j=1..k} x_j) for k>0.

Examples

			Square array begins:
   1,   1,    1,     1,      1,       1, ...
   3,   3,    3,     3,      3,       3, ...
   7,   9,   13,    21,     37,      69, ...
  15,  27,   63,   171,    495,    1467, ...
  31,  81,  321,  1521,   7761,   41361, ...
  63, 243, 1683, 14283, 131283, 1256283, ...
		

Crossrefs

Columns k=0-4 give: A000225(n+1), A000244, A001850, A206178, A216696.

Main diagonal gives A336204.

Cf. A309010.

Programs

Mathematica

T[n_, k_] := Sum[2^j * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)