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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 20*x^2 + 132*x^3 + 1140*x^4 + 11972*x^5 + 143156*x^6 +...
where we have the identities:
(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 +...
(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 +...
(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) +...
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PROG
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(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)}
for(n=0, 25, print1(a(n), ", "))
(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)}
for(n=0, 25, print1(a(n), ", "))
(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)}
for(n=0, 25, print1(a(n), ", "))
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