A360266 - OEIS

%I #19 Jul 13 2024 13:47:14

%S 1,2,6,22,82,312,1210,4752,18834,75184,301856,1217604,4930626,

%T 20032052,81615072,333328532,1364264250,5594210292,22977466864,

%U 94517423444,389316529512,1605533230256,6628467569292,27393187077144,113310732332274,469101108803052

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

%C Diagonal of rational function 1/(1 - (x + y + x^3*y^2)). - _Seiichi Manyama_, Mar 23 2023

%H Seiichi Manyama, <a href="/A360266/b360266.txt">Table of n, a(n) for n = 0..1000</a>

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

%F n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-3)*a(n-3).

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x^2)))

%Y Cf. A006139, A360267.

%Y Cf. A157004, A374598.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 31 2023