About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$

Question

For $n\ge 2\in\mathbb N$, let $S_n$ be the volume of a $(n-1)$ dimensional solid which satisfies $$\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n).$$

Then, here is my question.

Question : Can we represent $S_n$ by $n$ ?

Motivation : I've been interested in this simple question. I've got the followings :

$$S_2=2\sqrt 2, S_3=3\sqrt 3, S_4=\frac{32}{3}.$$

$S_4$ is the volume of a regular octahedron, whose edge length is $\sqrt 8$, which passes through the following six points :

$$(1,1,-1,-1),(1,-1,1,-1),(1,-1,-1,1),(-1,1,1,-1),(-1,1,-1,1),(-1,-1,1,1).$$

However, I don't have any good idea for $n$ in general. Can anyone help?

Update : I crossposted to MO.

Answer 1

I'm posting an answer just to inform that the question has received an answer by Igor Rivin on MO.