Closed form for $\prod_{1 \leq i < j \leq k} (j - i)$?

Question

Is there a closed form for $\prod_{1 \leq i < j \leq k} (j - i)$? It looks like something like a determinant of a Vandermonde matrix, but I can't seem to get it to fit.

Answer 1

Indeed, the square of this quantity is the discriminant of the polynomial whose roots are the integers from 1 to $k$, so your observation that this is the determinant of a Vandermonde matrix is correct. None of the below are close forms, but here are two alternative formulas that may (or may not) be helpful: $$\prod_{1\leq i < j \leq k}(j-i)=\prod_{n=1}^{k-1} n!=\prod_{n=1}^{k-1}n^{k-n}$$

Answer 2

You won't find a closed form, but you will find many references at http://oeis.org/A000178.