Let $A=1+\sum_{n=1}^\infty \alpha_nx^n\in\mathbb Z[[x]]$ and $B=\frac{1}{A}=1+\sum_{n=1}^\infty\beta_n x^n$ two mutually inverse power series having bounded integral coefficients (ie. $\vert \alpha_n\vert,\vert \beta_n\vert<C$ for some constant $C$ and for all $n$).

Examples are given by $A=\frac{P}{Q}$ where $P$ and $Q$ are both finite products of cyclotomic polynomials having only simple roots.

Are there other, exotic, examples?

More generally, consider again $A=1+\sum_{n=1}^\infty \alpha_nx^n\in\mathbb Z[[x]]$ with inverse $B=\frac{1}{A}=1+\sum_{n=1}^\infty\beta_n x^n$ and require that the integral coefficients $\alpha_n,\beta_n$ have at most polynomial growth (ie. there exists a constant $C$ such that $\vert\alpha_n\vert,\vert\beta_n\vert<Cn^C+C$ for all $n$).

Examples are now arbitrary rational fractions $A=\frac{P}{Q}$ involving only cyclotomic polynomials. Again, I know of no other examples. Do they exist?

Added: This question is closely linked to The sum of integers being a bijection, see Zaimi's comment after Venkataramana's anwser.