A333989 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1+(k-1)*x) / (1+2*(k-1)*x+((k+1)*x)^2).

1, 1, 1, 1, 0, 1, 1, -1, -4, 1, 1, -2, -7, 0, 1, 1, -3, -8, 23, 16, 1, 1, -4, -7, 64, 17, 0, 1, 1, -5, -4, 117, -128, -241, -64, 1, 1, -6, 1, 176, -527, -512, 329, 0, 1, 1, -7, 8, 235, -1264, 237, 4096, 1511, 256, 1, 1, -8, 17, 288, -2399, 3776, 11753, -8192, -5983, 0, 1

Offset: 0

Author

Seiichi Manyama, Sep 04 2020

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,  1,    1,    1,    1,     1, ...
  1,  0,   -1,   -2,   -3,    -4, ...
  1, -4,   -7,   -8,   -7,    -4, ...
  1,  0,   23,   64,  117,   176, ...
  1, 16,   17, -128, -527, -1264, ...
  1,  0, -241, -512,  237,  3776, ...
		

Crossrefs

Main diagonal gives A333991.

Cf. A307884, A333988.

Programs

Mathematica

T[n_, 0] := 1; T[n_, k_] := Sum[(-k)^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)

PARI

{T(n, k) = sum(j=0, n, (-k)^j*binomial(2*n, 2*j))}

Formula

T(n,k) = Sum_{j=0..n} (-k)^j * binomial(2*n,2*j).

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