Let $\lambda\in\mathbb{R}^n$ be a vector with positive components. Denote by $\vec{1}\in\mathbb{R}^n$ the vector with each component equal to $1$.
I am seeking to simplify the following sum for any fixed $i$:
$$ \sum_{\substack{G\in\{0,1\}^n\\G_i=1}} (-1)^{G^\top\vec{1}}\frac{\lambda_i}{G^\top\lambda}. $$
The sum is taken over all boolean vectors $G$ of size $n$ with the constraint that the $i$th component of $G$ must always be $1$. The $i$th component of $\lambda$ is denoted $\lambda_i$.
By the term "simplify", I mean find a way of computing it exactly with a computational complexity as low as possible. I am seeking a more efficient way of computing the sum than simply summing over all $2^{n-1}$ vectors $G$.