Does double subfactorial exist?

You will find a table of double subfactorials in sequence $A334578$ in $OEIS$ (have look here).

As you will read it in the title, they "simply" are $$!!n= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor }\,n\text{!!}\sum_{i=0}^{\left\lfloor \frac{n}{2}\right\rfloor}\frac{(-1)^i}{(n-2 i)\text{!!}}$$

Still according to the documentation, they obey the reccurence equation $$a_{n}=n\,a_{n-2}+(-1)^{\left\lfloor \frac{n}{2}\right\rfloor }\qquad \text{with} \qquad a_0=a_1=1$$

You will find a table fof the first $807$ terms (this is probably because this term is the last before $10^{1000}$).