Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form:

$$F=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M \space a_i \leq M^i$$.

and when the coefficients consist of a sequence of computably enumerable relation.

If the question is ambiguous ,please tell me but please do not downvote it.

When may Function (meromorphic) be expanded as power series with coefficients of integers