%I #12 Oct 14 2023 14:00:07
%S 1,1,0,1,4,6,8,29,84,162,360,1074,2808,6444,16464,45629,118244,297450,
%T 790184,2138438,5624136,14778068,39767024,107287122,286593800,
%U 768920084,2083170960,5642886852,15250029552,41369986008,112681853344,306930498205,836259756612
%N G.f. A(x) satisfies A(x) = 1 + x + x^3*A(x)^4.
%F a(n) = Sum_{k=0..floor(n/3)} binomial(3*k+1,n-3*k) * binomial(4*k,k)/(3*k+1).
%F a(n) = A366594(n) + A366594(n-1).
%o (PARI) a(n) = sum(k=0, n\3, binomial(3*k+1, n-3*k)*binomial(4*k, k)/(3*k+1));
%Y Cf. A137954, A366267, A366558.
%Y Cf. A366594.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Oct 13 2023