  We propose a class of generating functions denoted by $\textrm{RGF}_p(x)$,
which is related to the Sylvester denumerant for the quotients of numerical
semigroups. Using MacMahon's partition analysis, we can obtain
$\textrm{RGF}_p(x)$ by extracting the constant term of a rational function. We
use $\textrm{RGF}_p(x)$ to give a system of generators of the quotient of the
numerical semigroup $\langle a_1,a_2,a_3\rangle$ by $p$ for a small positive
integer $p$ and we characterise the generators for $\frac{\langle A\rangle}{p}$
for a general numerical semigroup $A$ and any positive integer $p$.

Combinatorics