A361726 Diagonal of rational function 1/(1 - (1 + x*y) * (x^2 + y^2)).

1, 0, 2, 4, 8, 24, 56, 144, 376, 960, 2512, 6560, 17184, 45248, 119296, 315392, 835552, 2217216, 5893568, 15687552, 41810944, 111567104, 298016512, 796832256, 2132456704, 5711486976, 15309014528, 41062927360, 110213725184, 295995574272, 795391639552

Offset: 0

Author

Seiichi Manyama, Mar 22 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A137635, A361727.

Cf. A115962, A360266.

Programs

PARI

a(n) = sum(k=0, n\2, binomial(2*k, k)*binomial(2*k, n-2*k));

Formula

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

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

a(n) ~ (1 + sqrt(3))^(n + 1/2) / (2*3^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 22 2023

n*a(n) = 2*(2*n-2)*a(n-2) + 4*(2*n-3)*a(n-3) + 2*(2*n-4)*a(n-4) for n > 3. - Seiichi Manyama, Mar 23 2023