Corollary

Let X be an indeterminate. The polynomials $(^X_n)$, where $n\in \mathbf{N}$, form a basis of the $\mathbf{Z}$-module consisting of the polynomials $P\in k[X]$ such that $P(\mathbf{Z})\subset \mathbf{Z}$.

If $P(\mathbf{Z})\subset \mathbf{Z}$, the Lagrange interpolation formula (Algebra, Chap. IV, §2, no. 1, Prop. 6) shows that the coefficients of P belong to $\mathbf{Q}$; thus, we can assume that $k=\mathbf{Q}$ and apply Prop. 2 with $V =\mathbf{Q},\mathscr{V}=\mathbf{Z}$.