Revista de la
Unión Matemática Argentina
On the restricted partition function via determinants with Bernoulli polynomials. II
Mircea Cimpoeaş
Volume 61, no. 2 (2020), pp. 431–440

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