A366589 G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)*A(x)^2.

1, 0, 0, 0, 1, 1, 0, 0, 2, 4, 2, 0, 5, 15, 15, 5, 14, 56, 84, 56, 56, 210, 420, 420, 342, 834, 1980, 2640, 2409, 3795, 9141, 15015, 16445, 20449, 43043, 80509, 104962, 123838, 215072, 419848, 630838, 780572, 1164228, 2190552, 3629704, 4884100, 6760390, 11715210

Offset: 0

Author

Seiichi Manyama, Oct 14 2023

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..4920

Crossrefs

Cf. A115178, A366588.

Cf. A366554.

Programs

Maple

f:= gfun:-rectoproc({(10 + 4*n)*a(n) + (26 + 8*n)*a(n + 1) + (16 + 4*n)*a(n + 2) + (-10 - n)*a(n + 5) + (-10 - n)*a(n + 6) = 0, a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1, a(5) = 1},a(n),remember):
map(f, [$0..30]); # Robert Israel, Oct 14 2024

PARI

a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(2*k, k)/(k+1));

Formula

G.f.: A(x) = 2 / (1+sqrt(1-4*x^4*(1+x))).

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(2*k,k)/(k+1).

(10 + 4*n)*a(n) + (26 + 8*n)*a(n + 1) + (16 + 4*n)*a(n + 2) + (-10 - n)*a(n + 5) + (-10 - n)*a(n + 6) = 0. - Robert Israel, Oct 14 2024