Is there a closed form expression for the following sum? $$\sum_{0\le i_1
I can understand that there are (nk) such terms and the values that i1+⋯+ik can take vary from k(k−1)2 to k(n−k)+k(k+1)2. So it remains to find how often the term ∑kj=1ij=l is found as an exponent in the above sum. Can anyone give some idea? Thanks.
Answer
This is a particular case of a more general formula that I posted in here Sum of power functions over a simplex . By denoting your sum as Sk(n) we have:
Sk(n)=k∑j=0(−1)jr12(−j+k−1)(k−j)+j(n+1)(r;r)j(r;r)k−j
where (r;r)j:=j−1∏l=0(1−rl+1).
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