# Singularity of Generating Function

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.

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Here is a pointer: generatingfunctionology. – Did May 8 '12 at 18:21
Lagrange reversion, contour integration, and analysis of poles seem very much on the analytic side to me... and gfology explains them all. Nevermind. The other classic book in this area is Analytic combinatorics, by Flajolet and Sedgewick. – Did May 8 '12 at 19:12
My last comment answered a comment by @Andy, now deleted. // On another note, there is a problem with the recursive relation on the coefficients $e_n$ since the only solution of the current one seems to be $e_n=0$ for every $n$. – Did May 8 '12 at 19:14
@Didier Thank you. I deleted my comment because I was not precise, there is a part of analytical comb. in Wilfs book. Analytic combinatorics, is the book I started studying. – Andy May 8 '12 at 19:19
@Didier $e_n$: I made a mistake in the sum, the highest index is now changed to be $n-1$ not $n$ – Andy May 8 '12 at 19:20

## 1 Answer

The relevant OEIS page, mentioned by @Diego in the comments, links to Analytic Combinatorics of Chord Diagrams by P. Flajolet and M. Noy, which refers to P.R. Stein and C.J. Everett, On a Class of Linked Diagrams II. Asymptotics, Discrete Mathematics 21 1978, 309-318, to assert that $$e_n\sim\frac1{\mathrm e}\frac{(2n)!}{2^nn!}.$$ Stirling's approximation yields $$e_n\sim\frac{\sqrt2}{\mathrm e}\left(\frac{2n}{\mathrm e}\right)^n,$$ hence the radius of convergence of the series $E(x)=\sum\limits_{n\geqslant0}e_nx^n$ is zero. The same is true for $F(x)$.

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