Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq n}(X_j-X_i)\right)^k,$$where $n,k$ are positive integers such that $n>1$ and $k(n-1)\equiv0 \pmod 2$.

Since $$\prod\limits_{1\leq i<j\leq n}(X_j-X_i)=\sum\limits_{\sigma\in S_n}sgn(\sigma)\prod\limits_{i=1}^nX_{\sigma_i}^{i-1},$$ here the sum is computed over all permutations $\sigma$ of the set $\{1,2,\cdots,n\}$, we can get the following simple results:

$(1)a_{n,1}=0$ for every odd positive integer $n>1$;

$(2)a_{n,2}\neq 0$ for every positive integer $n>1$.

I want to ask whether $a_{n,k}$ is equal to 0 or not when $k>2$.