OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
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
PROG
(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));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 06 2021
STATUS
approved