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A171214 G.f. A(x) satisfies A(x) = x + x*A(A(x/3)) = Sum_{n>=1} a(n)*x^n/3^(n*(n+1)/2).

%I A171214 #8 Feb 07 2025 06:46:58
%S A171214 1,1,2,10,137,5296,588365,190088818,179954321171,501722122937995,
%T A171214 4134242130461174144,100943613343624534183723,
%U A171214 7317423203727305175501741434,1577227642328692213664066391691150
%N A171214 G.f. A(x) satisfies A(x) = x + x*A(A(x/3)) = Sum_{n>=1} a(n)*x^n/3^(n*(n+1)/2).
%C A171214 More generally, if F(x) = x + x*F(F(qx)), then
%C A171214 F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx)) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
%C A171214 with a simple solution at q=1/2:
%C A171214 F(x) = x/(1-x/2) satisfies F(x) = x + x*F(F(x/2)).
%C A171214 At q=1, F(x,q=1) is the g.f. of A030266.
%C A171214 QUESTIONS regarding convergence of F(x,q) as a power series in x.
%C A171214 (1) What is Q, the maximum q below which a radius of convergence exists? Is Q=1?
%C A171214 (2) What is the radius of convergence for a given q < Q?
%e A171214 G.f.: A(x) = x + x^2/3 + 2*x^3/3^3 + 10*x^4/3^6 + 137*x^5/3^10 + 5296*x^6/3^15 +...+ a(n)*x^n/3^(n(n-1)/2) +...
%e A171214 A(x) = x + x*A(x/3) + x*A(x/3)*A(A(x/3)/3) + x*A(x/3)*A(A(x/3)/3)*A(A(A(x/3)/3)/3) +...
%e A171214 A(A(x)) = x + 2*x^2/3 + 10*x^3/3^3 + 137*x^4/3^6 + 5296*x^5/3^10 +...
%e A171214 SUMS OF SERIES at certain arguments.
%e A171214 A(1) = 1.423879975541542344910599787693637973194...
%e A171214 A(1/3) = 0.373293286580877833612329400906044642790...
%e A171214 A(A(1/3)) = A(1) - 1 = 0.42387997554...
%e A171214 A(A(1)) = 2.387414460111728675082753594461076041830...
%e A171214 A(3) = 3 + 3*A(A(1)) = 10.16224338033518602524826...
%o A171214 (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,x/3+O(x^n))));3^(n*(n-1)/2)*polcoeff(A,n)}
%Y A171214 Cf. A171212 (q=2), A171213 (q=3), A030266 (q=1).
%K A171214 nonn
%O A171214 1,3
%A A171214 _Paul D. Hanna_, Dec 08 2009