A signed-binary matrix is a matrix whose elements are either $-1$, $1$ or $0$.
A zero-sum matrix is a matrix such that
the sum of each column and each row is zero.
For example, $M=\begin{pmatrix}-1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & -1\end{pmatrix}$ is a zero-sum signed-binary matrix.
$a(n)$ is the number of zero-sum signed-binary matrices of size $n\times n$.
It is easy to see that $a(1)=1$ and $a(2)=3$.
Find $a(n)$.