Given
$E' = (E^2 + E - x)/2xE$
$xF = E^3 E' + 2xE^3 E'' + E^2 - x^2$
where $E = \sum_{n > 0}{e_n x^n}$
with $e_n = (n-1) \sum^{n-1}_{i = 1}{e_i e_{n-i}}$ for $n > 1$ and $e_1 = 1$
I am interested in finding the singularity of $F$ with smallest modulus, when interpreting $F$ as a function in the complex plane.
I just started studying analytic combinatorics by my self but my calculus knowledge is a bit rusty, so any pointers would be appreciated.