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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 12*x^2 + 69*x^3 + 531*x^4 + 4902*x^5 + 50829*x^6 +...
where we have the identities:
(0) A(x) = 1/(1-x) + (1+A(x))*x/(1-x)^2 + (1+A(x)^2)^2*x^2/(1-x)^3 + (1+A(x)^3)^3*x^3/(1-x)^4 + (1+A(x)^4)^4*x^4/(1-x)^5 + (1+A(x)^5)^5*x^5/(1-x)^6 +...
(1) A(x) = 1/(1-2*x) + A(x)*x/(1-x-x*A(x))^2 + A(x)^4*x^2/(1-x-x*A(x)^2)^3 + A(x)^9*x^3/(1-x-x*A(x)^3)^4 + A(x)^16*x^4/(1-x-x*A(x)^4)^5 + A(x)^25*x^5/(1-x-x*A(x)^5)^6 +...
(2) A(x) = 1 + x*(1 + (1+A(x))) + x^2*(1 + 2*(1+A(x)) + (1+A(x)^2)^2) + x^3*(1 + 3*(1+A(x)) + 3*(1+A(x)^2)^2 + (1+A(x)^3)^3) + x^4*(1 + 4*(1+A(x)) + 6*(1+A(x)^2)^2 + 4*(1+A(x)^3)^3 + (1+A(x)^4)^4) + x^5*(1 + 5*(1+A(x)) + 10*(1+A(x)^2)^2 + 10*(1+A(x)^3)^3 + 5*(1+A(x)^4)^4 + (1+A(x)^5)^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, (1 + A^m)^m * x^m / (1-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, A^(m^2) * x^m / (1-x - x*A^m +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) * (1 + A^k)^k +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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