Does there exist a non-trivial power series that equals its own coefficients on the positive integers?

Question

Consider the power series $p(x) = \sum_{n=0}^{\infty} a_n x^{n} $

Suppose this power series has infinite radius of convergence.

ie; $limsup_{n\rightarrow\infty} (|a_n|^{1/n}) = 0$

https://en.wikipedia.org/wiki/Power_series#Radius_of_convergence

Then can we have $p(x)$ such that $p(k) = a_k$ for all $k \in \mathbb{Z}^{+}$ ?

(There's obviously the trivial case where $a_{n} = 0$, but besides that?)

Given a nontrivial $p(x)$ exists, we should be able to multiply by any real to get at least continuum many such series, and there's only continuum many infinite sequences of real numbers, so we should have continuum many such series.

There's also continuum many convergent series that fail this condition (eg; series for nonzero constant real function).

Is there a nice way to map any such series (ie;series that fail to match their coefficients for positive integer x) to a series like $p(x)$? (or equivalently, map the relevant coefficient sequences to a $p(x)$ coefficient sequences)