On the Restricted Partition Function via Determinants with Bernoulli Polynomials

Let $$r\ge 1$$ be an integer, $${\mathbf {a}}=(a_1,\ldots ,a_r)$$ a vector of positive integers, and let $$D\ge 1$$ be a common multiple of $$a_1,\ldots ,a_r$$. We prove that, if a determinant $$\Delta _{r,D}$$, which depends only on r and D, with entries consisting in values of Bernoulli polynomials is nonzero, then 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 $$x_1\ge 0, \ldots , x_r\ge 0$$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.