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%I #15 Sep 24 2024 13:39:06
%S 1,0,0,1,1,0,5,11,6,35,120,136,336,1330,2310,4301,15456,35100,64701,
%T 193662,508921,1023000,2643432,7298984,16196682,38795055,105939288,
%U 254015541,596987183,1575487320,3959803694,9418896773,24081344034,61781452530,150293865540
%N G.f. satisfies A(x) = 1 + x^3*A(x)^5*(1 + x*A(x)).
%F a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+2*k+1,k) / (n+2*k+1).
%o (PARI) a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+2*k+1, k)/(n+2*k+1));
%Y Cf. A308616, A365723, A365724, A365725.
%Y Cf. A365695.
%K nonn
%O 0,7
%A _Seiichi Manyama_, Sep 17 2023