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%I #15 Feb 18 2023 02:26:54
%S 1,0,0,2,2,2,8,14,20,46,92,158,314,630,1176,2274,4498,8674,16804,
%T 32990,64358,125414,245832,481674,942912,1850122,3633220,7133730,
%U 14020694,27578954,54261912,106819006,210411028,414619486,817344908,1611978734,3180333830,6276743430
%N a(n) = Sum_{k=0..floor(n/3)} binomial(n-1-2*k,n-3*k) * binomial(2*k,k).
%F G.f.: 1 / sqrt(1-4*x^3/(1-x)).
%F n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n-3)*a(n-3) - 2*(2*n-6)*a(n-4).
%F a(n) ~ 2^(n-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 18 2023
%o (PARI) a(n) = sum(k=0, n\3, binomial(n-1-2*k, n-3*k)*binomial(2*k, k));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^3/(1-x)))
%Y Cf. A026585, A360310.
%Y Cf. A360291, A360314.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Feb 03 2023
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