Let $\mathbf s \in \mathbb N^N$ be a fixed vector, and define the polynomial $a(\cdot)$ such that \begin{equation} a(\mathbf x)=\prod_{i=1}^N\binom{x_i}{s_i} \end{equation} for all $\mathbf x \in \mathbb N^N$, where $\binom{x_i}{s_i} = \frac{x_i!}{s_i!(x_i - s_i)!}$.
I'd like to write this polynomial as a linear combination of monomials. In particular, I'd like to write \begin{equation} a(\mathbf x) = \sum_{\mathbf j \in \mathbb N^N} b_\mathbf j \mathbf x^\mathbf j, \end{equation} where $\mathbf x^\mathbf j = \prod_{i = 1}^N x_i^{j_i}$, and I have analytical expressions for the each coefficient $b_\mathbf j$ in terms of the vector $\mathbf s$.
Does anybody know how to do this?