OFFSET
0,3
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
T(n,k) = Sum_{j=0..n} (-k)^(n-j) * binomial(2*j,j) * binomial(2*n,2*j).
T(0,k) = 1, T(1,k) = 2-k and n * (2*n-1) * (4*n-5) * T(n,k) = (4*n-3) * (-4*(k-4)*n^2+6*(k-4)*n-k+6) * T(n-1,k) - (k+4)^2 * (n-1) * (2*n-3) * (4*n-1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 28 2020
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 1, 0, -1, -2, -3, ...
6, -5, -14, -21, -26, -29, ...
20, -41, -48, -7, 76, 195, ...
70, -125, 198, 739, 1222, 1395, ...
252, 131, 2080, 1629, -3772, -14873, ...
MATHEMATICA
T[n_, k_] := Sum[If[k == 0, Boole[n == j], (-k)^(n - j)] * Binomial[2*j, j] * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 27 2020 *)
PROG
(PARI) {T(n, k) = sum(j=0, n, (-k)^(n-j)*binomial(2*j, j)*binomial(2*n, 2*j))}
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Aug 27 2020
STATUS
approved