%I #7 Jan 19 2015 22:42:38
%S 1,4,20,132,1140,11972,143156,1874180,26280820,389655492,6059296564,
%T 98301356292,1657846584436,28995059053252,525023523671348,
%U 9831325298232068,190226868509537140,3800948683949700036,78387736603554834740,1667727317369199865604,36584269123749323010164
%N G.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n / (1-2*x)^(n+1).
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1-2*x - x*A(x)^n)^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * 2^(n-k) * (A(x)^k + 1)^k.
%e G.f.: A(x) = 1 + 4*x + 20*x^2 + 132*x^3 + 1140*x^4 + 11972*x^5 + 143156*x^6 +...
%e where we have the identities:
%e (0) A(x) = 1/(1-2*x) + (A(x)+1)*x/(1-2*x)^2 + (A(x)^2+1)^2*x^2/(1-2*x)^3 + (A(x)^3+1)^3*x^3/(1-2*x)^4 + (A(x)^4+1)^4*x^4/(1-2*x)^5 + (A(x)^5+1)^5*x^5/(1-2*x)^6 +...
%e (1) A(x) = 1/(1-3*x) + A(x)*x/(1-2*x - x*A(x))^2 + A(x)^4*x^2/(1-2*x - x*A(x)^2)^3 + A(x)^9*x^3/(1-2*x - x*A(x)^3)^4 + A(x)^16*x^4/(1-2*x - x*A(x)^4)^5 + A(x)^25*x^5/(1-2*x - x*A(x)^5)^6 + A(x)^36*x^6/(1-2*x - x*A(x)^6)^7 +...
%e (2) A(x) = 1 + x*(2 + (A(x)+1)) + x^2*(2^2 + 2*2*(A(x)+1) + (A(x)^2+1)^2) + x^3*(2^3 + 3*2^2*(A(x)+1) + 3*2*(A(x)^2+1)^2 + (A(x)^3+1)^3) + x^4*(2^4 + 4*2^3*(A(x)+1) + 6*2^2*(A(x)^2+1)^2 + 4*2*(A(x)^3+1)^3 + (A(x)^4+1)^4) + x^5*(2^5 + 5*2^4*(A(x)+1) + 10*2^3*(A(x)^2+1)^2 + 10*2^2*(A(x)^3+1)^3 + 5*2*(A(x)^4+1)^4 + (A(x)^5+1)^5) +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (A^m + 1)^m * x^m / (1-2*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * sum(k=0, m, binomial(m, k) * 2^(m-k) * (A^k + 1)^k +x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, A^(m^2) * x^m / (1-2*x - x*A^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A251177, A251178, A251180, A251181, A244610.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 19 2015