OFFSET
0,3
COMMENTS
In general, if g.f. satisfies: A(x)^2 = x + A(x*A(x)^q), q > 1, then a(n) ~ c(q) * q^n * n^(n - 1/q + (1/2 - 3/(2*q))*log(2)) / (exp(n) * log(2)^n), where c(q) is a constant independent on n.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
a(n) ~ c * 3^n * n^(n-1/3) / (exp(n) * (log(2))^n), where c = 0.3223127444389467551929... . - Vaclav Kotesovec, Aug 08 2014
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 233*x^4 + 4363*x^5 + 101905*x^6 +...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 38*x^3 + 504*x^4 + 9260*x^5 + 213757*x^6 +...
A(x*A(x)^3) = 1 + x + 5*x^2 + 38*x^3 + 504*x^4 + 9260*x^5 + 213757*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 64*x^3 + 819*x^4 + 14754*x^5 + 336467*x^6 +...
PROG
(PARI) {a(n)=local(A=[1, 1], Ax); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A]=Vec(1+subst(Ax, x, x*Ax^3) - Ax^2)[#A]); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 06 2014
STATUS
approved