Is there a rational function $F$ which may be expanded as power series with coefficients of unperiodical positive integers in such a form:

$$F(x)=\sum_0^{\infty}a_i x^i,\qquad a_i\in \mathcal{N} \cup \{0\},\quad\exists M\forall i,\space a_i \leq M.$$

Any unclear meaning may be clarified by comment and discussion.

Second question: Let $F$ be a function which may be expanded as power series with coefficients of unperiodical positive integers as above. Prove that $F$ is transcendental (I guess a function with expansion of power series with coefficients of bounded unperiodical positive integers has to be a rational function or transcendental).

Bull. SMF108(1980), 401–419): a formal series $f = \sum a_i x^i \in \mathbb{F}_q[[x]]$ with coefficients in a finite field (with $q=p^r$ elements) is algebraic if and only if its coefficient $a_i$ can be computed by a finite automaton from the base $p$ expansion of $i$. $\endgroup$ – Gro-Tsen Jun 2 at 10:04