This might be a question that shouldn't be asked here. But I need some help. I want to count the number of $n\times n$ symmetric matrices over the finite field $\mathbb{F}_q$ and rank $r$. I found the following note

http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf

But I think the formula given here is not correct. For example, it says that the number of symmetric matrices of rank $n$ is given by $$q^{{n \choose 2}}\prod\limits_{j=0}^s\left(1-(\frac{1}{q})^{2j-1}\right)$$

where $s=\lfloor{\frac{n}{2}}\rfloor$. But this is not matching with the simplest case, i.e. when $n=1$. Is there any other reference?