%I #21 Nov 15 2021 06:11:38
%S 1,3,9,34,147,684,3341,16896,87702,464566,2501178,13646625,75289022,
%T 419301351,2354121750,13309905653,75715795119,433063793430,
%U 2488921730886,14366319150072,83246947358766,484082947060300,2823980738817453,16522429720210884,96928401308507100
%N G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x)))^3.
%H Seiichi Manyama, <a href="/A349017/b349017.txt">Table of n, a(n) for n = 0..1000</a>
%F If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
%F a(n) ~ sqrt((1 - r*s)*(1 - r - r*s) / (1 - r*(s-1))) / (2*sqrt(Pi)*n^(3/2)* r^(n+1)), where r = 0.16019884639474132810520949540299923469792581229191347... and s = 2.80076422793129845097661115192234873280320027349745080... are real roots of the system of equations (-1 + r*s)^3/(-1 + r + r*s)^3 = s, (3*r^2*(-1 + r*s)^2)/(-1 + r + r*s)^4 = 1. - _Vaclav Kotesovec_, Nov 15 2021
%o (PARI) a(n, s=1, t=3) = sum(k=0, n, binomial(t*n-(t-1)*(k-1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));
%Y Cf. A262441, A349018.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Nov 06 2021