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.

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