OFFSET
0,13
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
FORMULA
A(n,k) is the coefficient of x^n in the expansion of (1 + x - k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, -1, -3, -5, -7, -9, -11, ...
1, -5, -11, -17, -23, -29, -35, ...
1, -5, 1, 19, 49, 91, 145, ...
1, 11, 81, 211, 401, 651, 961, ...
1, 41, 141, 181, 41, -399, -1259, ...
1, 29, -363, -2015, -5767, -12459, -22931, ...
MATHEMATICA
T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, May 02 2019
STATUS
approved