Is there any way to simplify $\frac{1}{2^{n - 1}}\sum_{i=0}^{n - 1} {\binom{n - 1}{i}}{a[i]}$?

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Here $a[i]$ denotes the i'th element (0 indexed) of a tuple. The goal is to avoid the huge coefficients since I know the final result won't be very big (in proportion to the elements of the tuple). I tried to use the following equation but I couldn't advance in this matter: $$\sum_{i=0}^{n} {\binom{n}{i}} = 2^{n}$$

$-1000 \leq a[i] \leq 1000, 0 \leq i < n$

$n \in \mathbb{N}$ , $a[i] \in \mathbb{R}$