On the restricted partition function

For a vector $\mathbf{a}=(a_1,\dots ,a_r)$ of positive integers we prove 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 $x_1\ge 0,\dots ,x_r\ge 0$ and its polynomial part.