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

## TL;DR

This paper extends previous work on the restricted partition function, showing it can be computed via linear systems involving Bernoulli polynomials and Barnes numbers when the common multiple is 1 or prime.

## Contribution

It proves that for certain values of D, the restricted partition function can be explicitly computed using Bernoulli polynomial-based linear systems.

## Key findings

- Valid for D=1 or prime D
- Expresses partition function via Bernoulli polynomials
- Provides explicit linear system formulation

## Abstract

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. In a continuation of a previous paper we prove that, if $D=1$ or $D$ is a prime number, the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ can be computed by solving a system of linear equations with coefficients which are values of Bernoulli polynomials and Bernoulli Barnes numbers.