n-"Kings" Problem

Question

I was thinking on some variations of the n-Queen problem until I reached the following problem I couldn't solve:
How many ways are there to put $n$ kings (as in a chess game) on a $n\times n$ chessboard such that no two kings threaten each other?
I'm not so optimistic that it would be solved in "non purely computational" methods. I've tried recursions, bijection and almost anything in my own knowledge yet I have achieved nothing.
Any help would be appreciated!

Answer 1

Based on the section 2.1.2 book "Non-attacking chess pieces" by "Václav Kotěšovec", the general answer does exist and is equal to the following : $$ \frac{n^{2n}}{n!}\left(1-\frac{9(n-1)n}{2n^2}+\frac{(n-1)n(243n^2-439n-142)}{24n^4}-\ldots\right) $$