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?