TITLE: Notation for factorial-type pattern with a skip/step of two instead of one?
QUESTION [5 upvotes]: I came across a peculiar pattern when solving a recurrence relation today:
Some sequence $a_n$ looks as such:
$a_0 = 1$
$a_2 = \frac{1}{2 \cdot 1}$
$a_4 = \frac{1}{4 \cdot 2 \cdot 1}$
$a_6 = \frac{1}{6 \cdot 4 \cdot 2 \cdot 1}$
...
The pattern is quite simple, but I can't think of a way to express the general equation for $a_n$
EDIT:
The recurrence relation is
$a_{n+2} = \frac{1}{n+2} \cdot a_n$
so the odd indices' denominators would be 1, 3, 5*3, 7*5*3, etc.
Any pointers?

REPLY [4 votes]: As mentioned, a slightly non-standard use of double-factorial gives
$$
a_n=\frac{1}{n!!}
$$
for even $n$. However, I usually see double factorial used with odd $n$. For even $n$,
$$
a_n=\frac{1}{2^{n/2}(n/2)!}
$$
works as well.
For odd $n$, without double factorial,
$$
a_n=\frac{2^{(n-1)/2}(\frac{n-1}{2})!}{n!}
$$
so it is easy to see why $1/n!!$ is preferred.