OFFSET
0,5
LINKS
Seiichi Manyama, Antidiagonals n = 0..50, flattened
FORMULA
T(n,k) = Sum_{j=0..floor((2*k-1)*n/(2*k))} (-1)^j * binomial(2*n,j) * binomial((2*k+1)*n-2*k*j-1,(2*k-1)*n-2*k*j) for k > 0.
EXAMPLE
(x^3 + x + 1/x + 1/x^3)^2 = x^6 + 2*x^4 + 3*x^2 + 4 + 3/x^2 + 2/x^4 + 1/x^6. So T(1,2) = 4.
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
0, 6, 44, 146, 344, 670, ...
0, 20, 580, 4332, 18152, 55252, ...
0, 70, 8092, 135954, 1012664, 4816030, ...
0, 252, 116304, 4395456, 58199208, 432457640, ...
MATHEMATICA
T[n_, 0] = Boole[n == 0]; T[n_, k_] := Sum[(-1)^j * Binomial[2*n, j] * Binomial[(2*k + 1)*n - 2*k*j - 1, (2*k - 1)*n - 2*k*j], {j, 0, Floor[(2*k - 1)*n/(2*k)]}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 02 2019
STATUS
approved