It is well known that the subring $S$ of integer valued polynomials ${\mathbb Q}[x]$ is generated by the binomial functions $P_n={x \choose n}$. One can ask a dual question: how to characterize the polynomial functions ${\mathbb Z} \to {\mathbb Z}$ which come from an element in ${\mathbb Z}[x]$. I understand one can write down derivative as a (terminating) series of difference derivatives and thus express each coefficients in terms of values of the polynomial but does this (or another) procedure lead to a neat answer?

There is a necessary condition for a polynomial function $f:{\mathbb Z} \to {\mathbb Z}$ to come from an element in ${\mathbb Z}[x]$, namely for every $n$ the residue of $f(x) \mod n$ depends only on the residue of $x \mod n$. This necessary condition is not sufficient but I am interested in the subring of elements in $S$ satisfying that necessary condition. Is there a nice set of generators and/or a basis?