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%I #5 Jan 19 2015 22:36:44
%S 1,1,2,11,59,376,2566,18646,141857,1120851,9141387,76635239,658411100,
%T 5784858465,51899580702,474971067333,4431203311040,42128438013171,
%U 408111843546201,4028707682556147,40534978365189110,415825232653264747,4350847058443120856,46450772334813948748
%N G.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n / (1+2*x)^(n+1).
%C Compare to: F(x) = Sum_{n>=0} (F(x)^n + 2)^n * x^n / (1+3*x)^(n+1) holds when F(x) = 1.
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1+2*x - 2*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 + 2)^k.
%e G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 59*x^4 + 376*x^5 + 2566*x^6 +...
%e where we have the identities:
%e (0) A(x) = 1/(1+2*x) + (A(x)+2)*x/(1+2*x)^2 + (A(x)^2+2)^2*x^2/(1+2*x)^3 + (A(x)^3+2)^3*x^3/(1+2*x)^4 + (A(x)^4+2)^4*x^4/(1+2*x)^5 + (A(x)^5+2)^5*x^5/(1+2*x)^6 +...
%e (1) A(x) = 1 + A(x)*x/(1+2*x - 2*x*A(x))^2 + A(x)^4*x^2/(1+2*x - 2*x*A(x)^2)^3 + A(x)^9*x^3/(1+2*x - 2*x*A(x)^3)^4 + A(x)^16*x^4/(1+2*x - 2*x*A(x)^4)^5 + A(x)^25*x^5/(1+2*x - 2*x*A(x)^5)^6 + A(x)^36*x^6/(1+2*x - 2*x*A(x)^6)^7 +...
%e (2) A(x) = 1 - x*(2 - (A(x)+2)) + x^2*(2^2 - 2*2*(A(x)+2) + (A(x)^2+2)^2) - x^3*(2^3 - 3*2^2*(A(x)+2) + 3*2*(A(x)^2+2)^2 - (A(x)^3+2)^3) + x^4*(2^4 - 4*2^3*(A(x)+2) + 6*2^2*(A(x)^2+2)^2 - 4*2*(A(x)^3+2)^3 + (A(x)^4+2)^4) - x^5*(2^5 - 5*2^4*(A(x)+2) + 10*2^3*(A(x)^2+2)^2 - 10*2^2*(A(x)^3+2)^3 + 5*2*(A(x)^4+2)^4 - (A(x)^5+2)^5) +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (A^m + 2)^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, A^(m^2) * x^m / (1+2*x - 2*x*A^m +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 + 2)^k +x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A251177, A251178, A251179, A251181, A244610.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 19 2015
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