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%I #19 Feb 27 2018 19:53:03
%S 1,2,7,26,130,764,4969,34586,253771,1944728,15462850,126936170,
%T 1071860068,9284365430,82326763672,746181872468,6905102562337,
%U 65185100425700,627340932873019,6152204560716188,61457807651698609,625212555146338418,6475800662211653437,68281122104004473972,732794914168656994990,8003359250151679857176
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (2 + 3*x*A(x)^n)^n.
%C Compare to the identity: if G(x) = Sum_{n>=0} x^n * (c + b*x*G(x)^n)^n,
%C then G(x) = Sum_{n>=0} b^n * x^(2*n) * G(x)^(n^2) / (1 - c*x*G(x)^n)^(n+1).
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} x^n * (2 + 3*x*A(x)^n)^n.
%F (2) A(x) = Sum_{n>=0} 3^n * x^(2*n) * A(x)^(n^2) / (1 - 2*x*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 26*x^3 + 130*x^4 + 764*x^5 + 4969*x^6 + 34586*x^7 + 253771*x^8 + 1944728*x^9 + 15462850*x^10 + ...
%e such that
%e A(x) = 1 + x*(2 + 3*x*A(x)) + x^2*(2 + 3*x*A(x)^2)^2 + x^3*(2 + 3*x*A(x)^3)^3 + x^4*(2 + 3*x*A(x)^4)^4 + x^5*(2 + 3*x*A(x)^5)^5 + x^6*(2 + 3*x*A(x)^6)^6 + ...
%e Also, the g.f. satisfies the identity:
%e A(x) = 1/(1 - 2*x) + 3*x^2*A(x)/(1 - 2*x*A(x))^2 + 3^2*x^4*A(x)^4/(1 - 2*x*A(x)^2)^3 + 3^3*x^6*A(x)^9/(1 - 2*x*A(x)^3)^4 + 3^4*x^8*A(x)^16/(1 - 2*x*A(x)^4)^5 + 3^5*x^10*A(x)^25/(1 - 2*x*A(x)^5)^6 + ...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(2 + 3*x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, 3^k*x^(2*k)*A^(k^2)/(1 - 2*x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A186998, A203014.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 25 2018
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