What is the number of $n \times n$ binary matrices with row and column sums 2 and with only zeros on the diagonal? This simple problem must have been treated somewhere, but I couldn't find any reference, even an OEIS entry. Among many aspects of this problem, I mostly care about the recurrence formula satisfied by enumeration result indexed by $n$. Note that this recurrence exists, as it can be shown that the sequence is P-recursive.

The case where the row and column sums all equal to 1 is the familiar problem of counting permutations with no fixed points. If the matrix is constrained to be symmetric, the result is given by the OEIS sequence A001205. It turns out that the non-symmetric case is quite more difficult.