I am trying to compute the number of distinct ways a $4n$ $\times$ $4n$ chessboard can be colored black and white, with exactly half the squares black and half the squares white. By distinct, I mean not equivalent by rotation or reflection, i.e. with respect to the dihedral group $D_4$.

Of course I wouldn't be too surprised if this has already been computed and the answer already known, but calculating a few terms and looking them up in OEIS didn't lead to anything. FYI, the first three terms are 1674, 229078019084673798, and 185026624806098273753009169783707528668060, corresponding to board sizes of 4$\times$4, 8$\times$8, and 12$\times$12.

Using Burnside and simplifying I've got it down to

$$\frac{1}{8} \left[2 {4 n^2 \choose 2 n^2}+3 {8 n^2 \choose 4 n^2}+{16 n^2 \choose 8 n^2} + 2 \sum _{k=0}^{2 n} {4 n \choose 2 k} {8 n^2-2 n \choose 4 n^2-k}\right]$$ and you can probably guess I would like a closed form for the sum.

With a little research and some help from Mathematica, the sum simplifies to

$$ \frac{(8n^2-2n)!}{(4n^2)!(4n^2-2n)!} \ {}_ {3}F_ 2\biggl(\begin{matrix} \frac{1}{2}-2n,\ -2n,\ -4n^2 \cr
\frac{1}{2},\ 4n^2-2n+1\end{matrix};-1\biggr)$$

I know little about hypergeometric functions but apparently some special cases are reducible to closed forms. My question is this, then: Does this particular hypergeometric expression have a closed form? And more generally, is there something akin to a table of integrals where I could look up something like this?

Clarification: Although my sequence does not appear in OEIS, it does appear as a *subsequence* of A082963, where two of three terms I computed are given. Unfortunately there is no formula given, closed or otherwise; nor is there any generating function mentioned, nor references cited. This is what I meant by my lookup in OEIS not leading to anything. Sorry to those I confused with my remark. The only reason I mentioned OEIS in the first place was to spare those kind enough to take interest in my question the time of looking there.