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%I #12 Apr 19 2024 05:00:48
%S 1,4,15,55,200,726,2640,9636,35343,130339,483395,1802901,6760781,
%T 25482643,96506229,367077447,1401772536,5372120718,20653929804,
%U 79634421312,307826528346,1192608522258,4629875048634,18006340509702,70142823370656,273633773330844
%N a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n+2,n-3*k).
%F a(n) = [x^n] 1/(((1-x)^3+x^3) * (1-x)^n).
%F a(n) = binomial(2*(1+n), n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1+n/3, (4+n)/3, (5+n)/3], 1). - _Stefano Spezia_, Apr 07 2024
%F a(n) ~ 2^(2*n+1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Apr 19 2024
%o (PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n+2, n-3*k));
%Y Cf. A001791, A120305, A371818, A371819.
%Y Cf. A371777.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Apr 06 2024
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