Let $n$ be a positive integer and $1\leq j\leq n$. Consider the following polynomial:

$$p_{n,j}(x)=\frac{\prod\limits_{i=1}^{n+1}\frac{x^{i}+1}{x+1}}{\prod\limits_{i=1}^{j}\frac{x^{i}+1}{x+1}\prod\limits_{i=1}^{n-j+1}\frac{x^{i}+1}{x+1}}\in\mathbb{F}[x]$$

This polynomial can be computed for each $j,n$ given. What I want to do is to understand in which cases $p(0)=0$ and in which cases $p(0)=1$ when working over a field $\mathbb{F}$ of characteristic $2$. I checked some cases by hand and this is what I got: $$p_{2,1}(0)=p_{2,2}(0)=1$$ $$p_{3,1}(0)=p_{3,2}(0)=p_{3,3}(0)=0$$ $$p_{4,1}(0)=p_{4,4}(0)=1~~p_{4,2}(0)=p_{4,3}(0)=0$$ $$p_{5,2}(0)=p_{5,4}(0)=1~~p_{5,1}(0)=p_{5,3}(0)=p_{5,5}=0$$ $$p_{6,1}(0)=p_{6,2}(0)=p_{6,3}(0)=p_{6,4}(0)=p_{6,5}(0)=p_{6,6}(0)=1$$ So apparently there is no easy pattern. It is also easily expressible as a product of cyclotomic polynomials, but I'm not sure if this is of any help. I would like (if possible) to have a way to know if $p_{n,j}(0)=0$ or $p_{n,j}(0)=1$ just in terms of $n$ and $j$.

If you are courious, these are the coefficients that has appeared in the computation of the homology groups of a certain family of Artin groups. I want to understand the behaviour of these polynomials over a field of characteristic $2$ in $x=0$ because this is the situation where I can assure that the homology groups are infinite dimensional.

Thanks for your help.