There is [this sequence](https://oeis.org/A002895) listed on OEIS - named *Domb numbers*. I'm curious about



> 
> **QUESTION.** Is there a direct combinatorial proof for the identity
> $$\sum\_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}
> =\sum\_{a+b+c+d=n}\binom{n}{a,b,c,d}^2,$$
> where the RHS is sum over all non-negative integers so that $a+b+c+d=n$.
> 
> 
> 



In search of a combinatorial proof for a multinomial sum
The [Richmond and Shallit paper](https://arxiv.org/pdf/0807.5028.pdf) linked from OEIS shows that both sides count the number of Abelian squares of length $2n$ on an alphabet of $4$ letters.


