Functions with integer-valued divided differences

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta < e^{1 + \tfrac{1}{2} + \cdots+ \tfrac{1}{m}} -1$.