Double derangement permutation

Question

Let $[n]$‎ ‌‎be a set with $n$ element‎. ‎A permutation $\sigma$ of the symmetric group $S_n$ is called a derangement of $[n]$ if $\sigma(i)\neq i$ for each $i\in[n]$‎. ‎Suppose that $\theta$ and $\gamma$ are two arbitrary permutations of $S_n$‎. ‎We say that a permutation $\sigma$ is a double derangement with respect to $\theta$ and $\gamma$ if $\sigma(i)\neq \gamma(i)$ and $\sigma(i)\neq \theta(i)$ for each $i\in[n]$ and denoted by $D_n(\gamma,\theta)$. Could you please give explicit formula or recurrence relation for the number of such derangement ($|D_n(\gamma,\theta)|$?