A336201 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, 0, 1, 1, -1, 0, 1, 1, -2, -3, 0, 1, 1, -3, -14, 11, 0, 1, 1, -4, -47, 136, 1, 0, 1, 1, -5, -134, 909, 106, -81, 0, 1, 1, -6, -347, 4736, 3585, -8492, 141, 0, 1, 1, -7, -846, 21655, 61906, -323523, 35344, 363, 0, 1, 1, -8, -1983, 91512, 771601, -8065624, 2201809, 395008, -1791, 0, 1

Offset: 0

Author

Seiichi Manyama, Jul 11 2020

Keywords

• sign
• 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, 0,  -1,    -2,      -3,       -4, ...
  1, 0,  -3,   -14,     -47,     -134, ...
  1, 0,  11,   136,     909,     4736, ...
  1, 0,   1,   106,    3585,    61906, ...
  1, 0, -81, -8492, -323523, -8065624, ...
		

Crossrefs

Columns k=0-3 give: A000012, A000007, (-1)^n*A098332(n), A336182.

Main diagonal gives A336202.

Cf. A309010, A336179, A336187.

Programs

Mathematica

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^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 *)