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.

I've scaled down the question from its earlier version to read as follows:

QUESTION. Let $\ell\geq 2$ be an integer. Are these integrals? $$\prod_{j=1}^n\frac{a_{2n}^3+a_{2n-2)}^3+\cdots+a_{2j}^3}{a_{2j}}.$$