The polynomial part of a restricted partition function related to the Frobenius problem (Q5942491)

Given a set \(A= \{a_1,\dots, a_m\}\) of positive integers, let \(p_A(t)\) denote the number of nonnegative integer solutions \((x_1,\dots, x_n)\) of the equation \(\sum a_ix_i= t\). Then \(p_A(t)\) can be written in the form \(\sum_\lambda P_{A,\lambda}(t)\lambda^t\), where the sum is over all complex numbers \(\lambda\) such that \(\lambda^{a_i}= 1\) for some \(i\), and where \(P_{A,\lambda}(t)\) is a polynomial in \(t\). The aim of this paper is to give an explicit formula for \(P_{A,1}(t)\); this is achieved in terms of the Bernoulli numbers.