I have a polynomial with integer coefficients on the form
$$ p_{l,N}(x) = x^{2l+1} - 2\sum_{m=0}^{l}\binom{2l+1}{2m}x^{2m}\sum_{j=1}^N j^{2l+1-2m} $$
and I am interested in finding the possible integer roots of this with $l\in\mathbb{N}:l\geq 1$ and $N\in\mathbb{N}:l\geq 1$. I think I was able to show that for specified values of $N$ and $l$, there is only one real root. In addition, if the root is integer, it must be an even number.
I guess one can use the integral root theorem to reduce the possible solutions to be factors of
$$2\sum_{j=1}^N j^{2l+1}$$
Am I right? If so, is there a way to find the factors of such a power sum in general? Thanks in advance for any help on this.