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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.

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I'm lazy to do this. Try to evaluate this expression for few values of $k$ say $k = 1,2,\ldots,10$. May be we can see a pattern? –  user2468 Feb 26 '12 at 20:48

2 Answers 2

up vote 1 down vote accepted

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: $$\prod_{1\leq i < j \leq k}(j-i)=\prod_{n=1}^{k-1} n!=\prod_{n=1}^{k-1}n^{k-n}$$

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Apparently there is not closed form for $\prod_{n=1}^{k-1} n!$ mathworld.wolfram.com/FactorialProducts.html –  user2468 Feb 26 '12 at 20:51
@J.D. So you say it's wrong (from there on)? –  draks ... Feb 26 '12 at 22:23
@draks No no! Indeed $\displaystyle\prod_{n=1}^{k-1} n!=\displaystyle\prod_{n=1}^{k-1}n^{k-n}$. I'm just answering OP's question whether a closed form exists. –  user2468 Feb 26 '12 at 22:53

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

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