Determinants with Bernoulli polynomials and the restricted partition function

Let $r\ge 1$ be an integer, $\mathbf{a}=(a_1,\dots ,a_r)$ a vector of positive integers and let $D\ge 1$ be a common multiple of $a_1,\dots ,a_r$. We study two natural determinants of order $rD$ with Bernoulli polynomials and we present connections with the restricted partition function $p_{\mathbf a}(n):=$thenumberofintegersolutions$(x_1,\dots,x_r)$to$\sum_{j=1}^r a_jx_j=n$with$x_1\geq 0, \ldots, x_r\geq 0$.