Prime Values of Double Factorial plus 1

Theorem
Let $n!!$ denote the double factorial function.

The sequence of positive integers $n$ such that $n!! + 1$ is prime begins:
 * $0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502, \ldots$

Proof
We have that: