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

If $n$ is odd. $n!!$ will only contain odd integers.

That means $n!!$ is odd. In the other hand, 1 is also an odd integer. We know that the sum two odd integers is always even.

Hence we can say that $n!! +1$ will be always even for all positive odd integers $n$ .

$n!!$ is known as the double factorial and is the product of every other integer decreasing all the way down to 1. For instance, $7!!=7\times5\times3\times1=105$ . For an odd integer $n$ , $n!!$ will always be a product of odd integers, and therefore odd itself; adding $1$ will always make an even integer.