I have two integer sequences $\{a\_n\}\_{n=0}^\infty$ and $\{b\_n\}\_{n=0}^\infty$. Explicit formulas for the $a\_n$ are known and their asymptotic growth is fully understood. My wish is to also understand the asymptotics of the numbers $b\_n$.


The corresponding power series $A(x) = \sum\_{n=0}^{\infty} a\_n x^n$ and $B(x) = \sum\_{n=0}^{\infty} b\_n x^n$ are related by an equation of the form


$$A(x) = B(p(x,A(x)))$$


where $p$ is a simple bivariate polynomial ($p(x,y) = xy^3$ in my concrete case).


I was able to find at least the exponential rate of growth of the $b\_n$ with my own pedestrian methods (essentially by determining the radius of convergence of $B(x)$). However, I was not able to determine the subexponential factor.


I have been looking for a general method in the "Analytic Combinatorics" book of Sedgewick and Flajolet, but did not find anything that seems to match my problem. I would appreciate any hints or pointers to solutions of similar problems.



Asymptotics of coefficients of implicitely defined generating function
Since $a\_0\neq 0$ there is (by the (formal) Lagrange theorem) a unique formal power series $x=x(z)$ solving $x={z \over A^3(x)}$, and $A(x(z))=B(z)$ because $A(x)=B(x\,A^3(x))$.
Expanding $A(x)$ using the Lagrange inversion formula gives $a\_0=b\_0$ and
$$b\_n=[z^n]A(x)={1 \over n}[t^{n-1}] A^\prime(t){1 \over A^{3n}(t)}=[t^n]{-1 \over 3n-1} {1 \over A^{3n-1}(t)}$$ for $n\geq 1$ (the latter equation because $[t^{n-1}]f^\prime(t)=n[t^n] f(t)$).  

The rhs is in many cases amenable to saddle point analysis, see e.g. section VIII.8 and Thm.VIII.8 in ''Analytic Combinatorics".


