Let $P(X)\in \mathbb Q[X]$ whereby $P(\lambda) \in \mathbb Z$ , $\forall \lambda \in \mathbb N$. Show that there are $a_{1},...,a_{r}$ $\in \mathbb Z$ and $n_{1},...,n_{r}$ $\in \mathbb N$ such that
$P(X)=a_{1}\binom{X}{n_{1}}+...+a_{r}\binom{X}{n_{r}}$
I am pretty stumped by the challenge question we received on our last question paper. I always find proofs to assert that there are such families $(a_{r})_{r}$ and $(n_{r})_{r}$ extremely difficult. Is induction even a starting point here given that $(a_{r})_{r} \in \mathbb Z$? How can we assert anything about the degree of the polynomial $P(X)$ without more information on the nature of $(n_{r})_{r}$?
Help is greatly appreciated.