For questions related to integer valued polynomials and their properties such as fixed divisor of a polynomial or representation in binomial base.
Integer valued polynomial is any polynomial $f(x)$ whose values $f(n)$ are integers for all integers $n$. Integer valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.
All integer valued polynomials can be written uniquely in form $$ f(x)=a_{n}\binom {x}{n}+a_{n-1}\binom {x}{n-1}+\dots+a_0\binom {x}{0}$$
Source: Integer-valued polynomial.