Determinant of $(p(x_i-x_j))_{1 \leq i,j \leq n}$ for polynomial $p(x)$.

Question

Let $p(x) = a_0 + a_1 x + \ldots + a_k x^k$ be a polynomial. Can anything be said in general about the determinant $\mathrm{det}_{1 \leq i,j \leq n} (p(x_i-x_j))$ for a collection of variables $x_1,\ldots,x_n$?

It seems that for small $n,k$, the first order term in the determinant - which is itself a polynomial in the variables $x_1,\ldots,x_n$ - is a Vandermonde determinant.