On the restricted partition function via determinants with Bernoulli polynomials. II

Abstract

Let $r \geq 1$ be an integer, $a = (a_1,...,a_r)$ a vector of positive integers, and let $D \geq 1$ be a common multiple of $a_1,...,a_r$. We prove that if $D=1$ or $D$ is a prime number then the restricted partition function $p_a(n) := $ the number of integer solutions $(x_1,...,x_r)$ to $∑_{j=1}^r a_j x_j=n$, with $x_1 \geq 0,..., x_r \geq 0$, can be computed by solving a system of linear equations with coefficients that are values of Bernoulli polynomials and Bernoulli–Barnes numbers.