%I #6 Mar 30 2012 18:37:27
%S 1,2,8,34,140,586,2476,10522,45048,194210,842672,3678946,16155140,
%T 71328210,316536532,1411398138,6321140080,28426660498,128325523272,
%U 581349815466,2642337533500,12046547596514,55076433751372,252470682559914
%N G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * 2*x^(2*n-1)/(1 - 2*x^(2*n-1)).
%C Related q-series identity:
%C Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)) = Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)); here q=x, y=A(x), z=2.
%F G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2^n*A(x)*x^n/(1 - A(x)*x^(2*n)).
%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 34*x^3 + 140*x^4 + 586*x^5 + 2476*x^6 +...
%e which satisfies the following relations:
%e A(x) = 1 + A(x)*2*x/(1-2*x) + A(x)^2*2*x^3/(1-2*x^3) + A(x)^3*2*x^5/(1-2*x^5) +...
%e A(x) = 1 + 2*A(x)*x/(1-A(x)*x^2) + 4*A(x)*x^2/(1-A(x)*x^4) + 8*A(x)*x^3/(1-A(x)*x^6) +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,A^m*2*x^(2*m-1)/(1-2*x^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,2^m*A*x^m/(1-A*x^(2*m)+x*O(x^n))));polcoeff(A,n)}
%Y Cf. A192400, A192403.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 30 2011