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A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2*n) / Product_{k=1..n} (1 - x^k*A(x))^2.

%I A206639 #10 Mar 30 2012 18:37:35
%S A206639 1,1,4,18,91,489,2751,15985,95218,578324,3568084,22299964,140885754,
%T A206639 898292262,5772951668,37355908797,243184468271,1591567315702,
%U A206639 10465836784159,69114490893596,458171948148640,3047865264442504,20339282134624054,136122586785459512
%N A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2*n) / Product_{k=1..n} (1 - x^k*A(x))^2.
%F A206639 G.f. satisfies the identities:
%F A206639 (1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n+1) / Product_{k=1..n} (1 - x^k*A(x)).
%F A206639 (2) A(x) = 1/(1 - Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1 - x^k*A(x)) ).
%e A206639 G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 91*x^4 + 489*x^5 + 2751*x^6 +...
%e A206639 where the g.f. satisfies:
%e A206639 (0) A(x) = 1 + x*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^4/((1-x*A(x))^2*(1-x^2*A(x))^2) + x^9*A(x)^6/((1-x*A(x))^2*(1-x^2*A(x))^2*(1-x^3*A(x))^2) +...
%e A206639 (1) A(x) = 1 + x*A(x)^2/(1-x*A(x)) + x^2*A(x)^3/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)^4/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
%o A206639 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, 1-x^k*A+x*O(x^n))^2)); polcoeff(A, n)}
%o A206639 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(m+1)/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
%o A206639 for(n=0,30,print1(a(n),", "))
%Y A206639 Cf. A196150, A145268, A206637.
%K A206639 nonn
%O A206639 0,3
%A A206639 _Paul D. Hanna_, Feb 11 2012