It is well known that the sum of squares of all binomial coefficients ${n\choose k}$ with a fixed $n$ is ${2n\choose n}$. That is, $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$.

But do we know what the value of the sum of squares of multinomial coefficients is? In particular, I am interested in $\sum_{a,b,c,d} {n\choose a,b,c,d}^2$, where the sum is taken over all 4-tuples ${a,b,c,d}$ of nonnegative integer whose sum is $n$.

Thanks.