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%I #11 Sep 17 2023 10:10:59
%S 1,0,0,1,1,0,4,9,5,22,78,91,175,680,1224,1938,6270,14630,24794,63756,
%T 166980,322920,720720,1900080,4125888,8803008,22151360,51778804,
%U 111882100,267682272,645736432,1442390092,3346519020,8094247798,18657762006,42890295734
%N G.f. satisfies A(x) = 1 + x^3*A(x)^4*(1 + x*A(x)).
%F a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+k+1,k) / (n+k+1).
%o (PARI) a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+k+1, k)/(n+k+1));
%Y Cf. A308616, A365723, A365724, A365726.
%Y Cf. A054514.
%K nonn
%O 0,7
%A _Seiichi Manyama_, Sep 17 2023
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