OFFSET
0,5
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
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).
EXAMPLE
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, ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jan 19 2020
STATUS
approved