Variable cv.complexe – Is there a function that can be extended in infinite series of powers with bounded positive coefficients?

Let F be a rational function, is there such an F that can be extended as a series of powers with coefficients of non-periodic positive integers of the following form:

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

Any obscure meaning can be clarified by comments and discussions