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A192259 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^n * A(x)^(n*(n+1)/2).

%I A192259 #9 Jun 20 2016 22:45:18
%S A192259 1,2,10,74,658,6514,69210,773306,8974114,107288162,1314003882,
%T A192259 16420439978,208754062258,2693915486418,35228738082298,
%U A192259 466239274517274,6238546207411778,84330947396776642,1150982783030893386,15854319075541606666,220344302315492953298,3089322686040279975474,43693043476823499717018,63085549664634453982706,6423320378114329801258421518738
%N A192259 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^n * A(x)^(n*(n+1)/2).
%F A192259 Let A = g.f. A(x), then A satisfies:
%F A192259 (1) A = Sum_{n>=0} x^n*(1+A)^n*A^n * Product_{k=1..n} (1 - x*(1+A)*A^(2*k-1))/(1 - x*(1+A)*A^(2*k))
%F A192259 (2) A = 1/(1- A*(1+A)*x/(1- A*(A-1)*(1+A)*x/(1- A^3*(1+A)*x/(1- A^2*(A^2-1)*(1+A)*x/(1- A^5*(1+A)*x/(1- A^3*(A^3-1)*(1+A)*x/(1- A^7*(1+A)*x/(1- A^4*(A^4-1)*(1+A)*x/(1- ...))))))))) (continued fraction)
%F A192259 The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression.
%e A192259 G.f.: A(x) = 1 + 2*x + 10*x^2 + 74*x^3 + 658*x^4 + 6514*x^5 +...
%e A192259 Let A = g.f. A(x), then A satisfies:
%e A192259 A = 1 + x*(1+A)*A + x^2*(1+A)^2*A^3 + x^3*(1+A)^3*A^6 + x^4*(1+A)^4*A^10 +...
%e A192259 Equivalently,
%e A192259 A = 1 + x*(A + A^2) + x^2*(A^3 + 2*A^4 + A^5) + x^3*(A^6 + 3*A^7 + 3*A^8 + A^9) +...
%o A192259 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(1+A)^m*(A+x*O(x^n))^(m*(m+1)/2)));polcoeff(A,n)}
%o A192259 for(n=0,30,print1(a(n),", "))
%Y A192259 Cf. A107591, A192260.
%K A192259 nonn
%O A192259 0,2
%A A192259 _Paul D. Hanna_, Jun 26 2011