I am looking for a simple closed-form expression simplifying the sum (or average, if you prefer) of the geometric means of all subsamples $A\subset X$ of cardinality $|A|=k$ from a grand set $X$, \begin{equation*} \sum_{A\subset X: |A|=k}\left(\prod_{x\in A} x \right)^{1/|A|} \end{equation*} Is there any identity that would allow for such simplification?
The question is motivated by the expression \begin{equation*} \sum_{e\in\{0,1\}^N:\sum_i e_i=k}\exp\left(\sum_{i=1}^N x_i e_i \right) \end{equation*} which I am hoping to simplify.
