The so-called *$\ell$-sequences* are defined by $a_0=0, a_1=1$ and $a_n=\ell\,a_{n-1}-a_{n-2}$. The *Generalized Lecture Hall Theorem* (due to Mireille BousquetMelou and Kimmo Eriksson) depends on a polynomial analogue of $\ell$-sequences.

>**QUESTION.** Let $\ell\geq 2$ and $r, k\geq1$ be integers. Are these integrals?
$$\prod_{j=1}^n\frac{a_{rn}^{2k-1}+a_{r(n-1)}^{2k-1}+\cdots+a_{rj}^{2k-1}}{a_{rj}}.$$