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COMMENTS
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This is an application of the more general formula given by:
if G(x) = Series_Reversion(x - x*F(x)), with F(0)=0, then
(1) G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*F(x)^n/n!,
(2) G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*F(x)^n/n! );
here F(x) = A(x) and G(x) = A(x) - x^2.
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PROG
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(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x^2+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x+x^2); for(i=1, n, A=x+x^2+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x+x^2); for(i=1, n, A=x^2+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
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