I was reading about integer-valued-polynomials -- rational single-variable polynomials that evaluate to integers when integers are plugged in for the variable. I read that all binomial polynomials $\binom{x}{n}$ = $\frac{x (x-1) \cdots (x-n+1)}{n!}$ are integer valued polynomials.
Similarly, I wanted to see (even examples would suffice) that evaluate to integers when n! is plugged in (for an natural 'n') but wouldn't evaluate to integers for all integers 'n'. A concrete example apart from above mentioned binomial polynomials or any source that talks about it.
A simple example is $$f(x)=\frac{x(x-1)(x-2)(x-4)}{5}.$$ For $n\in\Bbb Z$, $f(n)$ is an integer iff $n\not\equiv 3\pmod 5$, and there is no factorial $\equiv3\pmod 5$.