OFFSET
0,9
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
A(n,k) is the constant term in the expansion of (k/x + 1 + k*x)^n.
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..n} (1-2*k)^(n-j) * k^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (1+2*k)^(n-j) * (-k)^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k^2) * (n-1) * A(n-2,k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 9, 19, 33, 51, ...
1, 7, 25, 55, 97, 151, ...
1, 19, 145, 595, 1729, 4051, ...
1, 51, 561, 2611, 8001, 19251, ...
1, 141, 2841, 22141, 105441, 369501, ...
1, 393, 12489, 119449, 627873, 2319801, ...
MATHEMATICA
T[n_, k_] := Sum[If[k == j == 0, 1, k^(2*j)] * Binomial[n, 2*j] * Binomial[2*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
nonn,tabl
AUTHOR
Seiichi Manyama, May 01 2019
STATUS
approved