Thursday, 13 October 2016

combinatorics - Is there a closed form expression for the following sum?



Is there a closed form expression for the following sum? $$\sum_{0\le i_1


I can understand that there are \binom{n}{k} such terms and the values that i_1+\cdots+i_k can take vary from \frac{k(k-1)}{2} to k(n-k)+\frac{k(k+1)}{2}. So it remains to find how often the term \sum_{j=1}^k i_j=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 S_k(n) we have:
\begin{equation} S_k(n) = \sum\limits_{j=0}^{k} \frac{(-1)^j r^{\frac{1}{2} (-j+k-1) (k-j)+j (n+1)}}{(r;r)_j (r;r)_{k-j}} \end{equation}
where (r;r)_j := \prod\limits_{l=0}^{j-1} (1-r^{l+1}).


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