Double Factorial

On Day 24771, my tweet was:

24771: 3×23×359; member of OEIS A259359: numbers n such that n!!-8 is prime (useful as a means of generating probable primes) #mathematics. 

Prior to this I'd been perplexed until I read the following in Wikipedia:

The double factorial should not be confused with the factorial function iterated twice, which is written as (n!)! and not n!!

This of course is quite confusing. The Wikipedia goes on to say that:

The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as
1, 2, 8, 48, 384, 3840, 46080, 645120,... (sequence A000165 in the OEIS) 

The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as
1, 3, 15, 105, 945, 10395, 135135,... (sequence A001147 in the OEIS) 

Perhaps the alternative term semifactorial is more appropriate and odd factorial is sometimes used for the double factorial of an odd number. Interestingly, the Wikipedia article mentions that double factorials also arise in expressing the volume of a hypersphere. The 4-dimensional hypersphere, or 3-sphere, was the subject of my previous post. The formula for the volume of an n-dimensional hypersphere is:

 

which reduces to the familiar formula for the volume of a sphere when n=3.