Given $r_1,\ldots,r_n\in\mathbb{R}$, the objective is to compute the following summation: \begin{align} \sum_{\substack{x_1,\ldots,x_n\in \{0,1\}\\x_1+\ldots+x_n<n/2}} {r_1}^{x_1}\cdots{r_n}^{x_n}, \end{align} i.e., low order terms of $(r_1+1)\ldots(r_n+1)$. Summation in exhaustive manner requires exponential complexity in $n$. I guess that the summation can be simplified if indices have a particular structure. For instance, say $x_1+\ldots+x_n=0 \ (\text{mod } 2)$, then the summation boils down to \begin{align} \frac{1}{2}\left((r_1+1)\ldots(r_n+1)-(r_1-1)\ldots(r_n-1)\right). \end{align} Is there any efficient summation method for my problem?
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When you say efficient, do you mean you want something that fits in at most a few pages of handwritten work, or do you mean a good way for a computer to do it with some tolerance? – Calvin Khor Sep 29 '20 at 01:51
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@CalvinKhor I mean the latter one. – Heedong Do Sep 29 '20 at 01:55