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?
> $$\prod\_{j=1}^n\frac{a\_{2n}^3+a\_{2n-2}^3+\cdots+a\_{2j}^3}{a\_{2j}}.$$
> 
> 
> 



Integrality of ratios of $\ell$-sequences
Yes, this is true for any odd exponent $2k-1$ on place of 3.


First of all, $a\_m$ is a monic polynomial in $\ell$ of degree $m-1$, and these polynomials are known as Chebyshev polynomials of second kind: if $\ell=2\cos x$, then $a\_m=\frac{\sin mx}{\sin x}$. So it suffices to prove that $A:=\prod\_{j=1}^n (a\_j^{2k-1}+\ldots+a\_n^{2k-1})$ is divisible by $B:=a\_1\ldots a\_n$ as a polynomial (then the ratio is a monic polynomial in $\ell$ with integer coefficients). The roots of $a\_m$ are simple and are equal to $2\cos \pi t/m$, $t=1,\ldots,m$. So, if $0<u<v$ are integers and $u,v$ are coprime, the number $2\cos \pi u/v$ is a root of $B$ of multiplicity $\lfloor n/v\rfloor:=q$. Thus it suffices to prove that there exist at least $q$ indexes $j$ for which $a\_j^{2k-1}+\ldots+a\_n^{2k-1}$ is divisible by $a\_v$, call these indexes *good*. Consider $q$ sets $\Delta\_i:=\{iv,iv-1,\ldots,(i-1)v+1\}$, $i=1,\ldots,q$. I claim that each $\Delta\_i$ contains a good index $j$. Denote $n=qv+r$, $0\leqslant r<v$. Look at the remainders of $a\_i$'s modulo $a\_v$. They are $0=a\_0,a\_1,a\_2,\ldots,a\_{v-1},0$, then $-a\_{v-1},-a\_{v-2},\ldots,-a\_1,0$, then $a\_1,a\_2,\ldots$ etc. Thus $a\_{qv+j}\equiv -a\_{qv-j} \pmod {a\_v}$ for all $j=0,1,\ldots$ that implies that $qv-r\in \Delta\_q$ is good, so is $n-2v\in \Delta\_{q-1}$, $qv-r-2v\in \Delta\_{q-2}$ etc.


