TITLE: Non unique factorization of integer valued polynomials
QUESTION [6 upvotes]: Is there a nice example of a polynomial with non unique factorization in the subring of 
$\mathbb Q[X,Y]$ of polynomials that defines functions $\mathbb Z^2\to\mathbb Z$?
I don't think this subring is a UFD because of the possibility to multiply the unique factors in $\mathbb Q[X,Y]$ in different ways to sometimes obtain different irreducible factors in the subring.

REPLY [4 votes]: Worth emphasis is how extremely different the lengths of irreducible factorizations can be in such nonunique factorizations in the ring of integer-valued polynomials. For example
$$ n {x\choose n}\, =\, (x-n+1){x\choose n-1}$$
The RHS is a product of two irreducibles, but the LHS can have an arbitrarily large number of irreducible factors by choosing $n$ with many prime factors.
For completeness, below is a proof of the irreducibility of $x\choose n$ from the 2016 Monthly paper What You Should Know About Integer-Valued Polynomials by Cahen and Chabert.