Related? https://artofproblemsolving.com/community/c6h419576p2369305

Using the above result, one can prove that there exist at least $(n+1)-\lceil (n-1)/2\rceil$ non-zero coefficients. For even $n$, the equality case holds when, for example, $a_k=(-1)^k\lceil k/2\rceil$ for $k=1,2,\dotsc ,n$. For odd $n$, I have not yet find an example.

For $n = 2k+1$, we have $P(x) = \prod_{i = -k}^k (x-i) + \varepsilon$, for a small enough value of $\varepsilon$.