I am not considering $(n!)!$ .

Number Theory Level 2

$1!! + 1$ is even.

$3!! +1$ is even.

$5!! + 1$ is even.

$7!! +1$ is even.

$9!! +1$ is even.

Is it true that $n!! + 1$ is always an even number for all positive odd integers $n?$

I am not considering (n!)!(n!)!(n!)!.