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%I #10 Sep 17 2023 10:11:06
%S 1,0,0,1,1,0,2,5,3,5,21,28,26,84,180,207,385,990,1562,2288,5278,10439,
%T 15925,30212,64738,112268,192780,396338,761634,1317840,2512940,
%U 5015867,9146049,16729462,32927950,62851525,115101637,220051350,427898900,800956600,1505524800
%N G.f. satisfies A(x) = 1 + x^3*A(x)^2*(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, A365724, A365725, A365726.
%Y Cf. A365694.
%K nonn
%O 0,7
%A _Seiichi Manyama_, Sep 17 2023
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