In how many ways can we fill $n \times n$ matrix with $+1$ or $-1$ such that the product of the entries in each row and each column equals $-1$?
Let the matrix size be $n$.
For each row and each column, there must be an odd number of $(-1)$s. (Why?)
You can arbitrarily assign $1$ and $-1$ for each element of the upper-left $(n-1)\times(n-1)$ cells. The last column will have to balance its previous rows (aka make the row product $-1$), and the last row will have to balance the previous columns (aka make the column product $-1$).
Can you derive a summation expression now?