A361753 a(n) = Sum_{k=0..floor(n/3)} binomial(2*(n-3*k),k) * binomial(2*(n-3*k),n-3*k).

1, 2, 6, 20, 74, 276, 1044, 3994, 15426, 60008, 234764, 922716, 3640700, 14411952, 57210750, 227659704, 907853778, 3627085932, 14515139376, 58174092472, 233463067284, 938061587212, 3773298437204, 15193083455580, 61230698571372, 246978403761112

Offset: 0

Author

Seiichi Manyama, Mar 23 2023

Keywords

• nonn

Comments

Diagonal of rational function 1/(1 - (1 + (x*y)^3) * (x + y)).

Links

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

Crossrefs

Cf. A137635, A361752.

Cf. A360267.

Programs

PARI

a(n) = sum(k=0, n\3, binomial(2*(n-3*k), k)*binomial(2*(n-3*k), n-3*k));

Python

from math import comb
def A361753(n): return sum(comb(m:=(r:=n-3*k)<<1,k)*comb(m,r) for k in range(n//3+1)) # Chai Wah Wu, Mar 23 2023

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^3)^2).

Recurrence: n*a(n) = 2*(2*n-1)*a(n-1) + 8*(n-2)*a(n-4) + 2*(2*n-7)*a(n-7). - Vaclav Kotesovec, Mar 23 2023