Triple factorial in terms of simple factorial

Question

For the double factorial and even numbers $(2k)$ is well known that: $$(2k)!!=2^k k!$$ Which is easy to see just from the definition. From here we can get for odd numbers $(2k-1)$: $$(2k-1)!!=\dfrac{(2k)!}{2^k k!}.$$ Then we can express double factorial in terms of simple factorials and powers of $2$. I wonder if there is some well known formula for triple factorial (or multifactorial in general). Is always easy to see that for $3k$: $$(3k)!!!=3^k k!$$ But I don't have idea of how to find a formula for $3k-1$ or $3k-2$: $$(3k-1)!!!=?$$ $$(3k-2)!!!=?$$ I also would love to see if there is a general way of find these formulas for any multifactorial. Thanks.

Answer 1

Unfortunately, the examples you mentioned are the only "nice extensions" for multifactorials. Anything else requires expansions with the Gamma function, which this answer and the corresponding thread explain pretty well.

As you would expect, separating double factorials by parity allowed for the simplifications above, but triple factorials and higher level multifactorials lack this capability.