A349017 G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)))^3.
1, 3, 9, 34, 147, 684, 3341, 16896, 87702, 464566, 2501178, 13646625, 75289022, 419301351, 2354121750, 13309905653, 75715795119, 433063793430, 2488921730886, 14366319150072, 83246947358766, 484082947060300, 2823980738817453, 16522429720210884, 96928401308507100
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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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));
Formula
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).
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
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A300048. - Seiichi Manyama, Dec 04 2024