OFFSET
0,9
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).
D-finite with recurrence: 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, 3, 5, 7, 9, 11, 13, ...
1, 7, 13, 19, 25, 31, 37, ...
1, 19, 49, 91, 145, 211, 289, ...
1, 51, 161, 331, 561, 851, 1201, ...
1, 141, 581, 1441, 2841, 4901, 7741, ...
1, 393, 2045, 5797, 12489, 22961, 38053, ...
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
Main diagonal gives A187018.
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, May 01 2019
STATUS
approved