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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x - x*A'(x)*A(x) ) = x.
(2) A(x) = x + A(x) * A'(A(x)) * A(A(x)).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * A'(x)^n * A(x)^n / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * A'(x)^n * A(x)^n / n! ).
a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 1.5447806483693... and c = 0.02888888614196289496..., conjecture: alfa = 2*(2*LambertW(1) - 1 + 1/(1 + LambertW(1))). - Vaclav Kotesovec, Feb 22 2023
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EXAMPLE
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G.f.: A(x) = x + x^2 + 5*x^3 + 42*x^4 + 471*x^5 + 6422*x^6 + 101439*x^7 + 1803949*x^8 + 35459549*x^9 + 760744865*x^10 + ...
such that A( x - x*A'(x)*A(x) ) = x.
Related series.
Series_Reversion(A(x)) = x - x^2 - 3*x^3 - 22*x^4 - 235*x^5 - 3153*x^6 - 49721*x^7 - 888784*x^8 - 17615520*x^9 + ...
A'(x)*A(x) = x + 3*x^2 + 22*x^3 + 235*x^4 + 3153*x^5 + 49721*x^6 + 888784*x^7 + 17615520*x^8 + ...
A(A(x)) = x + 2*x^2 + 12*x^3 + 110*x^4 + 1294*x^5 + 18127*x^6 + 290620*x^7 + 5206800*x^8 + 102633591*x^9 + ...
A'(A(x)) = 1 + 2*x + 17*x^2 + 208*x^3 + 3108*x^4 + 53328*x^5 + 1018948*x^6 + 21297818*x^7 + 481458997*x^8 + ...
A'(A(x))*A(A(x)) = x + 4*x^2 + 33*x^3 + 376*x^4 + 5242*x^5 + 84625*x^6 + 1534652*x^7 + 30682881*x^8 + ...
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A=serreverse(x - x*A'*A +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, x^m*(A')^m*A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(A')^m*A^m/m!)) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
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