It is well-known (using for example the Vandermonde's convolution identity) that $$\sum\limits_{j=0}^n{n \choose j}^2={2n \choose n}.$$ During my calculation I got the following sum $$\sum\limits_{k_1+...+k_n=k}{k \choose k_1,...,k_n}^2,$$ with ${k \choose k_1,...,k_n}$ being the multinomial coefficients.

Is the value of this sum known in literature? (I didn't find neither any formula which can allow for computation of it nor the combinatorial proof.) Any hints?


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