The double factorial notation
The way I parse it, $n!$ is a post-fix function call. Therefore $n!!$ parses as
factorial(factorial(n)). This notation has my disapprobation.
Well, the way I assume it was chosen was:
With one factorial sign, we decrement by one:
$n! = n \cdot (n - 1) \cdot ... \cdot 2 \cdot 1$
So, if we have two factorial signs, let's decrement by two!
$n!! = n \cdot (n - 2) \cdot ... \cdot 3 \cdot 1$
And thus, it was done.
It's a terrible abuse of notation, as mentioned in the other answer, I'll agree.