%I #12 Aug 08 2014 09:45:39
%S 1,1,2,21,340,7459,203191,6563082,244275977,10276093457,481793838446,
%T 24906702642493,1407549620790557,86342073783944457,
%U 5714570300215357125,405973384652872885758,30817592926484298640320,2489727584432844543366345,213310909894977172829098783
%N G.f. satisfies: A(x)^3 = x + A(x*A(x)^2)^2.
%H Vaclav Kotesovec, <a href="/A240997/b240997.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ c * 2^n * n^(n - 1/3 - log(2/3)/4) / (exp(n) * (log(3/2))^n), where c = 0.253289930600610020471... . - _Vaclav Kotesovec_, Aug 08 2014
%e G.f.: A(x) = 1 + x + 2*x^2 + 21*x^3 + 340*x^4 + 7459*x^5 + 203191*x^6 +...
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 5*x^2 + 46*x^3 + 726*x^4 + 15682*x^5 + 423101*x^6 +...
%e A(x*A(x)^2) = 1 + x + 4*x^2 + 34*x^3 + 540*x^4 + 11696*x^5 + 316071*x^6 +...
%e A(x*A(x)^2)^2 = 1 + 2*x + 9*x^2 + 76*x^3 + 1164*x^4 + 24744*x^5 + 661010*x^6 +...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 76*x^3 + 1164*x^4 + 24744*x^5 + 661010*x^6 +...
%o (PARI) {a(n)=local(A=[1,1],Ax);for(i=1,n,A=concat(A,0);Ax=Ser(A);
%o A[#A]=Vec(1+subst(Ax^2,x,x*Ax^2) - Ax^3)[#A]);A[n+1]}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A240996.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 06 2014