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? $$\prod_{j=1}^n\frac{a_{2n}^3+a_{2n-2)}^3+\cdots+a_{2j}^3}{a_{2j}}.$$

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? $$\prod_{j=1}^n\frac{a_{2n}^3+a_{2n-2}^3+\cdots+a_{2j}^3}{a_{2j}}.$$

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? $$\prod_{j=1}^n\frac{a_{rn}^{2k-1}+a_{r(n-1)}^{2k-1}+\cdots+a_{rj}^{2k-1}}{a_{rj}}.$$

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? $$\prod_{j=1}^n\frac{a_{2n}^3+a_{2n-2)}^3+\cdots+a_{2j}^3}{a_{2j}}.$$