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EXAMPLE
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E.g.f.: A(x) = x + 8*x^2/2! + 204*x^3/3! + 8656*x^4/4! + 514100*x^5/5! +...
where A(5*x - 4*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 8*x^2 + 204*x^3 + 8656*x^4 + 514100*x^5 + 39254904*x^6 +...
where F(x)/x = 1/5 + 4/(5-x)^2 + 4^2/(5-2*x)^3 + 4^3/(5-3*x)^4 + 4^4/(5-4*x)^5 +...
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PROG
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(PARI) {a(n) = local(A=x); A = serreverse(5*x - 4*x*exp(x +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x+sum(m=1, n, Dx(m-1, 4^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 4^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
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