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+1,2*j).
T(0,k) = 1, T(1,k) = 6-k and n * (2*n+1) * (4*n-3) * T(n,k) = (4*n-1) * (-4*(k-4)*n^2+2*(k-4)*n+k-2) * T(n-1,k) - (k+4)^2 * (n-1) * (2*n-1) * (4*n+1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 29 2020
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
6, 5, 4, 3, 2, 1, ...
30, 11, -6, -21, -34, -45, ...
140, -29, -120, -139, -92, 15, ...
630, -365, -266, 531, 1654, 2755, ...
2772, -1409, 2520, 6489, 4828, -5853, ...
MATHEMATICA
T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * Binomial[2*j, j] * Binomial[2*n+1, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)
PROG
(PARI) {T(n, k) = sum(j=0, n, (-k)^(n-j)*binomial(2*j, j)*binomial(2*n+1, 2*j))}
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Aug 28 2020
STATUS
approved