Let $$r\_k(x)=\prod\_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$


Computations suggest that $$r\_{2k}(x)=\sum\_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r\_{2k+1}(x)=\sum\_{j=0}^{k^2-k}a(2k+1,j)\binom{k^2+k+x-j}{k^2+k},$$ where the coefficients $a(k,j)$ are positive, palindromic and gamma positive.
Is there an elementary proof?



Worpitzky-like identities?
Upgrading my comment to an answer. It is not hard to see that $r\_k(x)$ is the same as MacMahon's famous formula for the number of [plane partitions](https://en.wikipedia.org/wiki/Plane_partition) in a $\lfloor k/2 \rfloor \times \lceil k/2 \rceil \times x$ box. Then the same argument as in [my previous answer](https://mathoverflow.net/questions/402767/generating-functions-for-hankel-determinants-of-catalan-numbers/402795#402795) (see in particular Section 3.15.2 of Stanley's EC1, which explains that $a(k,j)$ is the number of linear extensions of $[\lfloor k/2 \rfloor]\times [\lceil k/2 \rceil]$ with $j$ descents) implies the $\gamma$-positivity you are interested in.


