How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$?

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$ so that each column and each row has exactly $n $ $1$'s and $n$ $-1$'s ?

I tried for cases $n=1 , 2$ but the solutions were just case checking so I can't apply it to the general case.

This question was asked and answered on Math Overflow. For reasons of completeness and convenience, I have copied and collated the information that was given there.

As stated by others, this sequence can be found in OEIS A058527 which lists the first 7 terms.

An explicit (closed) formula for this was published about 30 years ago, but it was wrong. As the matter stands, there is no explicit formula. The values up to $m=15$ can be found here, and the values relevant to this specific question are all those of the form $B(2m,m; 2m,m)$.

For convenience, I have listed them below.

$$\begin{align} m=1. \quad & 2 \\ m=2. \quad & 90 \\ m=3. \quad & 297200 \\ m=4. \quad & 116963796250 \\ m=5. \quad & 6736218287430460752 \\ m=6. \quad & 64051375889927380035549804336 \\ m=7. \quad & 108738182111446498614705217754614976371200 \\ m=8. \quad & 34812290428176298285394893936773707951192224124239796250\\ m=9. \quad & 21882630320667689225357109687240364487595251549773\ 48944382853301460850000 \end{align}$$

$m=10.$ 27856726619148825494784078580561902698029561072477313921903857594739813713335069572835719440 ,

$m=11.$ 73598361490497320315061303329860141625489297388873097402577973995078475595177219907273524251166598220695445665280 ,

$m=12.$ 41166741168628144936713296088571382312319541605035039202139739426526710130553316487181153798718623434772470791046740497537457960747230000 ,

$m=13.$ 4955718943988940005004146237479895414479770954795218135246063162883554097069851160345377950138482450273137136644609946481127918748277794634799510166778789400000000 ,

$m=14.$ 130182965915524188822374132662598705148400755946368804279519820149265310510762181776582751351898520706779949869976891542842075030612982451615770950941110943399182350404672558394208550600000000 ,

$m=15.$ 755108153829514055973248475938007693494252921038915181695050280737084462727343843042892520013526446320417069829825720805149300044864243469236136642966675916041398513473858832514945641793476366441713887508829265486123827200 ,