Both $\{\frac{x^n}{n!}:n\in\mathbb{N}\}$ and $\{\frac{x^n}{n^2+1}:n\in\mathbb{N}\}$ are Schauder bases for the ring of formal power series $\mathbb{R}[[x]]$ as a topological vector space over $(\mathbb{R},discrete)$. But they differ in one respect. Let us say that a Schauder basis $\{B_n:n\in\mathbb{N}\}$ for $\mathbb{R}[[x]]$ has the multiplication property if for all sequences $(a_n)$ and $(b_n)$ of nonnegative integers, if $(\Sigma_n a_n B_n)(\Sigma_n b_n B_n) = \Sigma_n c_n B_n$ then $(c_n)$ is a sequence of nonnegative integers. Then $\{\frac{x^n}{n!}:n\in\mathbb{N}\}$ has the multiplication property, while $\{\frac{x^n}{n^2+1}:n\in\mathbb{N}\}$ does not. That explains why $\Sigma_n a_n \frac{x^n}{n!} $ is a useful generating function for combinatorics, whereas $\Sigma_n a_n \frac{x^n}{n^2+1}$ is not.
My question is, is there a way to characterize which Schauder bases for $\mathbb{R}[[x]]$ have the multiplication property?
In general, given $\{f_n(x)\}$ is a Schauder basis for $\mathbb{R}[[x]]$, $\{f_n(x)\}$ has the multiplicative property if and only if every product $f_n(x)f_m(x)$ is a linear combination of $f_k(x)$ with non-negative integer coefficients.
Looking at Schauder bases with vectors of the form $f_k(x)=a_kx^k$, this condition translates to saying that $a_ma_n$ is an integral multiple of $a_{m+n}$ for all natural numbers $m,n$. In particular, if $a_i=1$ for some $i$, then $a_m$ is an integral multiple of $a_{m+i}$, which can prove useful in evaluating if a Schauder basis of this form has the multiplicative property if one keeps note of all the terms which have coefficient 1.
This is probably not the most efficient answer to the problem, as this condition might be difficult to verify for more complicated Schauder bases like Hermite polynomials. Nevertheless, hope you found this answer helpful.