It is possible to show using diverse techniques that the following polynomial:

$$P_n(x)=1 + \binom{n}{2} x + \binom{n}{4} x^2 + \binom{n}{6} x^3 + \binom{n}{8} x^4 +\ldots + \binom{n}{2\lfloor\tfrac{n}{2}\rfloor} x^{\lfloor \frac{n}{2}\rfloor},$$

is real-rooted. For instance, it is an $s$-Eulerian polynomial, for which Savage and and Visontai in [\[1\]][1] have proved real-rootedness.

Here I ask for a very similar polynomial, which I think might be solved with some other technique that perhaps I am overlooking.

$$Q_n(x)=1 + \left(\binom{n}{2}-n\right) x + \binom{n}{4} x^2 + \binom{n}{6} x^3 + \binom{n}{8} x^4 +\ldots + \binom{n}{2\lfloor\tfrac{n}{2}\rfloor} x^{\lfloor \frac{n}{2}\rfloor},$$

This polynomial $Q_n(x) = P_n(x) - nx$ is the Ehrhart $h^*$-polynomial of the hypersimplex $\Delta_{2,n}$. There are several conjectures regarding unimodality/log-concavity/real-rootedness for the $h^*$-polynomial of an arbitrary hypersimplex $\Delta_{k,n}$, but for $k > 2$ the formulas are much more complicated.



  [1]: https://arxiv.org/pdf/1208.3831.pdf