Hello.

The notation means that for each tuple (n_1,n_2,...,n_k) of non-negative integers, such that n_1+n_2+...+n_k=n, we get a term in the sum.

The order of the terms in the sum does not matter, but if it makes it any clearer, here are some examples of what the tuple (n_1,n_2,...,n_k) might be:

(n,0,0,...,0),

(n-1,1,0,0,...,0),

(n-2,1,1,0,0,...,0),

...

(n-2,2,0,0,...0),

...

(0,0,...,0,n).

So for every such tuple, you get a term on the left hand side.