We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, denoted by $s(X')$. What is a simplified formula of the product of the individual summations of all nonempty subsets of the set $X$?
For example, let $X$ contain $\{1, 2, 3\}$. Our nonempty subsets $X'$ are $\{1\}$, $\{2\}$, $\{3\}$, $\{1, 2\}$, $\{1, 3\}$, $\{2, 3\}$, $\{1, 2, 3\}$, and their summations $s$ are $1, 2, 3, 3, 4, 5,$ and $6,$ respectively. We wish to find a simplified way of representing their product $(1)(2)(3)(3)(4)(5)(6)$ without having to multiply it all out.