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.

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