Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by $$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$ Notice that $P_n(2)=n$ for every $n\in\mathbb{Z}_{\geq0}$. Here are a few examples: $$P_2=x, \qquad P_3=x^2-1, \qquad P_4=x^3-2x, \qquad P_5=x^4-3x^2+1.$$

QUESTION 1.Empirical evidence suggest that, for each fixed integers $n, k\geq1$, $$Q_{n,k}(x):=\frac{P_1(x)^{2k-1}+P_2(x)^{2k-1}+\cdots+P_n(x)^{2k-1}}{P_1(x)+P_2(x)+\cdots+P_n(x)} \tag1$$ is a polynomial in $x$.This is trivial for $k=1$. Is it true for other odd powers $2k-1$?

**REMARKS.**

(1) Specialized values $2k-1=3$ or $5$, etc are still interesting to me.

(2) Even the case of special valuations for $x\in\mathbb{Z}$ are appealing as well, which means (1) becomes a claim on integrality of sequences.

QUESTION 2.Encouraged by the success with QUESTION 1, how about this? $$R_n(x)=\prod_{j=1}^n\frac{P_j(x)^{2k-1}+\cdots+P_n(x)^{2k-1}}{P_j(x)} \tag2$$ is a polynomial in $x$.