Any algebraic function that can be expanded as infinite series with unbounded positive integral coefficients?

Question

Cound one give examples of irrational algebraic function that can be expanded as infinite series with unbounded positive integral coefficients, like:

$$f(x)=\sum_0^{\infty}a_i x^i,\qquad a_i\in \mathbb{N} \cup \{0\}$$

I post the question for interest, since the first time it has not been clearly expressed, I have updated the question.

Answer 1

$$\frac{1 - \sqrt{1 - 4x}}{2x} = \sum_{k=0}^\infty C_n x^n$$ where $C_n$ are the Catalan numbers.

There are many other examples in the OEIS.