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a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).
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%I #13 Feb 03 2023 01:37:44

%S 1,2,6,20,70,254,936,3492,13150,49882,190318,729576,2807816,10841962,

%T 41983588,162973568,633994982,2471010742,9646981054,37718873700,

%U 147676286078,578883674722,2271704404900,8923807316892,35087269756344,138075819924306

%N a(n) = Sum_{k=0..floor(n/4)} binomial(n-1-3*k,k) * binomial(2*n-8*k,n-4*k).

%H Seiichi Manyama, <a href="/A360292/b360292.txt">Table of n, a(n) for n = 0..1000</a>

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

%F n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-4)*a(n-4) - 2*(2*n-13)*a(n-5) - (n-8)*a(n-8).

%o (PARI) a(n) = sum(k=0, n\4, binomial(n-1-3*k, k)*binomial(2*n-8*k, n-4*k));

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

%Y Cf. A085362, A360290, A360291.

%Y Cf. A360295.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 01 2023