The Polynomial Part of a Restricted Partition Function Related to the Frobenius Problem

Given a set of positive integers $ A = \{ a_{1} , \dots , a_{n} \} $, we study the number $ p_{A} (t) $ of nonnegative integer solutions $ \left( m_{1} , \dots , m_{n} \right) $ to $ \sum_{j=1}^{n} m_{j} a_{j} = t $. We derive an explicit formula for the polynomial part of $p_A$.