Consider a taylor series with nonzero radius :
$$f_c(x) = \sum_{k = 0}^{\infty} a_k x^k =a_0 + a_1x + a_2x^2 + ...$$
Such that the set/list $a_n$ is a bijection to the set of integers larger than a given integer $c$.
There are no restrictions on $a_n$ such as increasing or order etc but remember the radius has to be positive.
( In fact I think the radius is always $1$ because of absolute convergeance and the fact that polynomials grow slower than exponentials, however I am uncertain due to the maybe weird order the integers are put into .. and also poles, singularities etc etc. But that is not the main point here )
The question is :
Does such an $f_c(x)$ (with positive radius) exist for some $c$ such that
$f_c(x)$ is not a rational function.
$f_c(x)$ has a closed form.
I have not formally defined " closed form " and I intend to have a very open mind for it.
One of the reasons I tend not to consider this likely to exist is for instance
I believe
$f_c(x)^j =\sum_{k = 0}^{\infty} j_k x^k =j_0 + j_1x + j_2x^2 + ...$ for integer $j$ has coefficients much more complicated than the set of integers without repetition, either by size or repetition. And hence it would surprise me to see a closed form or cancellation with other closed form functions of $f_c(x)$.
Similarly
The integral or derivative of $f_c(x)$ are very different when one considers the set of taylor coefficients. Again it would surprise me to see closed forms or cancellation with other closed forms.
Note that imo we can probably make this function a lacunary series by for instance putting the relatively large integers near $x^{n^4}$ and the smaller integers more equally distributed with the other powers. But lacunary is also not a closed form usually and not a rational function.
One more note :
I avoided the idea of the set $\frac{1}{m}$ ( the inverse of the nonzero integers, including negatives ) because of ideas like Riemann Series Theorem.
https://en.wikipedia.org/wiki/Riemann_series_theorem
How to prove it ?
I got insprired from this one
Does there exist a generating function for the rational numbers?
( and I am skeptical there too )