We consider the asymtotics of the coefficients of  generating function $y(x)$, 
which satisfies the implicit function $y= F(x,y)$.

If $F(x,y)$ is a rational function in $y$ and has non-negative coefficients of development in $x$ and $y$, 
can we conculde that $[x^n]y(x)\sim \rho^{-n} n^{-\frac{3}{2}}$? 

If not, what kind of additional condition do we need to prove such asymtotics?

It seems that it can be derived from singularity analysis. A reference could be 
Flajolet and Sedgewick's Book  "Analytic Combinatorics". However the proof in there is rather obscure for me. 

I think that we still need to additional condition to make that conclusion,  such as there exist $r>0$ and $s>0$ satisfying
\begin{equation}
F(r,s)=s, F_x(r,s)=1.
\end{equation}


Any reference about this topic would be appreciated. Thank You in Advance.