On the restricted partition function II

Let $\mathbf{a}=(a_1,\dots ,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other formulas for the restricted partition function $p_{\mathbf{a}}(n):=$ the number of integer solutions $(x_1,\dots ,x_r)$ to ${\sum }_{j=1}^r{a}_j{x}_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.