  We investigate multi-graded Gorenstein semigroup algebras associated with an
infinite family of reflexive lattice simplices. For each of these algebras, we
prove that their multigraded Poincar\'e series is rational. Our method of proof
is to produce for each algebra an explicit minimal free resolution of the
ground field, in which the resolution reflects the recursive structure encoded
in the denominator of the finely-graded Poincar\'e series. Using this
resolution, we show that these algebras are not Koszul, and therefore
rationality is non-trivial. Our results demonstrate how interactions between
multivariate and univariate rational generating functions can create subtle
complications when attempting to use rational Poincar\'e series to inform the
construction of minimal resolutions.
