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A192401 G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * x^n/(1 - x^(2*n)).

%I A192401 #6 Mar 30 2012 18:37:27
%S A192401 1,1,2,6,16,49,156,512,1728,5959,20886,74204,266624,967141,3536814,
%T A192401 13025478,48266972,179831935,673258626,2531481990,9555606112,
%U A192401 36196916933,137554950152,524265889839,2003513188296,7675473295796,29471911733772
%N A192401 G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * x^n/(1 - x^(2*n)).
%C A192401 Related q-series identity:
%C A192401 Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=x, y=1, z=A(x).
%F A192401 G.f. satisfies: A(x) = 1 + Sum_{n>=1} A(x)*x^(2*n-1)/(1 - A(x)*x^(2*n-1)).
%e A192401 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 49*x^5 + 156*x^6 +...
%e A192401 which satisfies the following relations:
%e A192401 A(x) = 1 + A(x)*x/(1-x^2) + A(x)^2*x^2/(1-x^4) + A(x)^3*x^3/(1-x^6) +...
%e A192401 A(x) = 1 + A(x)*x/(1-A(x)*x) + A(x)*x^3/(1-A(x)*x^3) + A(x)*x^5/(1-A(x)*x^5) +...
%o A192401 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,A^m*x^m/(1-x^(2*m)+x*O(x^n))));polcoeff(A,n)}
%o A192401 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,A*x^(2*m-1)/(1-A*x^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
%Y A192401 Cf. A192400, A192403.
%K A192401 nonn
%O A192401 0,3
%A A192401 _Paul D. Hanna_, Jun 30 2011