A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).

1, 1, 0, 1, 3, -6, 1, 6, 6, 0, 1, 9, 30, 10, 30, 1, 12, 66, 140, 15, 0, 1, 15, 114, 450, 630, 21, -140, 1, 18, 174, 1000, 2955, 2772, 28, 0, 1, 21, 246, 1850, 8430, 18963, 12012, 36, 630, 1, 24, 330, 3060, 18855, 69384, 119812, 51480, 45, 0

Offset: 0

Author

Seiichi Manyama, Jan 19 2020

Keywords

• sign
• tabl

Examples

			Square array begins:
    1,  1,    1,     1,     1,      1, ...
    0,  3,    6,     9,    12,     15, ...
   -6,  6,   30,    66,   114,    174, ...
    0, 10,  140,   450,  1000,   1850, ...
   30, 15,  630,  2955,  8430,  18855, ...
    0, 21, 2772, 18963, 69384, 187425, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=1..5 give A000217(n+1), A002457, A002695(n+1), A331515, A331516.

Cf. A307883, A331511.

Programs

Mathematica

T[n_, k_] = 1/2 * Sum[If[k == 2 && n == j - 1, 1, (k - 2)^(n + 1 - j)] * j * Binomial[n + 1, j] * Binomial[n + 1 + j, j], {j, 1, n + 1}]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)

PARI

T(n,k) = (1/2)*sum(j=1,n+1,(k-2)^(n+1-j)*j*binomial(n+1,j)*binomial(n+1+j,j));
matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jan 20 2020

Formula

T(n,k) = (1/2) * Sum_{j=1..n+1} (k-2)^(n+1-j) * j * binomial(n+1,j) * binomial(n+1+j,j).

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

T(n,k) = ((n+2)/2) * Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).

T(n,k) = Sum_{j=0..n} (k/2)^j * (-(k-2)^2/(2*k))^(n-j) * (2*j+1) * binomial(2*j,j) * binomial(j,n-j) for k > 0. - Seiichi Manyama, Aug 20 2025