TITLE: The coefficient of a specific monomial in the expansion of the following polynomial
QUESTION [6 upvotes]: 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 $k>2$.

REPLY [10 votes]: This is not an answer but a conjecture based on some trials.
$$a_{n,k} = \begin{cases} 1, & n=1 \\
                          0, & n\gt 1, k\text{ odd} \\
                          (-1)^{t\binom n2}\displaystyle\frac{(tn)!}{(t!)^n}, & k=2t.
            \end{cases}
$$
Note that $\frac{(tn)!}{(t!)^n}$ is the number of ways of partitioning a set of size $tn$ into $n$ sets of size $t$.
Oh, now I see that this is a special case of Macdonald's constant-term conjecture, see this article, which has references.