How to write sums of combinations of signed sums? I am looking for a short way to write sums of combinations of $x_1,x_2,\ldots,x_n$ for a practical application. It can be  easier explained using an example:
For $n=3$ there are $2^3=8$ combinations to sum $x_1,x_2,x_3$ using all signed combinations: $$C_3=-f\left(-x_1-x_2-x_3\right)+f\left(x_1-x_2-x_3\right)+f\left(-x_1+x_2-x_3\right)-f\left(x_1+x_2-x_3\right)+f\left(-x_1-x_2+x_3\right)-f\left(x_1-x_2+x_3\right)-f\left(-x_1+x_2+x_3\right)+f\left(x_1+x_2+x_3\right)$$
Every combination are summed arguments of a function $f$ and has a different sign sequence.  The signs of $f$ are negative/positive if the number of negative signs in the arguments are odd/even.
For $n=4$ there are already $2^4=16$ combinations and it is quite ineffective to write them all down:
$$C_4=f\left(-x_1-x_2-x_3-x_4\right)-f\left(x_1-x_2-x_3-x_4\right)-f\left(-x_1+x_2-x_3-x_4\right)+f\left(x_1+x_2-x_3-x_4\right)-f\left(-x_1-x_2+x_3-x_4\right)+f\left(x_1-x_2+x_3-x_4\right)+f\left(-x_1+x_2+x_3-x_4\right)-f\left(x_1+x_2+x_3-x_4\right)-f\left(-x_1-x_2-x_3+x_4\right)+f\left(x_1-x_2-x_3+x_4\right)+f\left(-x_1+x_2-x_3+x_4\right)-f\left(x_1+x_2-x_3+x_4\right)+f\left(-x_1-x_2+x_3+x_4\right)-f\left(x_1-x_2+x_3+x_4\right)-f\left(-x_1+x_2+x_3+x_4\right)+f\left(x_1+x_2+x_3+x_4\right)$$
It is clear that for higher $n$ a short notation is needed. I do not look for a program code.
 A: We can write $C_n$, for every non-negative integer $n$, as $$C_n = \sum_{(s_1, s_2, \dots, s_n) \in \{-1, 1\}^n}s_1s_2\cdots s_n \cdot f\Big(s_1x_1 + s_2x_2 + \dots + s_nx_n\Big),$$
where $\{-1, 1\}^n := \Big\{(s_1, s_2, \dots, s_n) \mid s_i \in \{-1, 1\} \text{ for all positive integers } i \leq n\Big\}$.
The idea is to just use the "variables" $s_i$ to act as the signs.
For example, $\{-1, 1\}^3$ is defined to be
$$\{(\color{red}{-1}, \color{red}{-1}, \color{red}{-1}), (\color{red}{-1}, \color{red}{-1}, \color{green}{+1}), (\color{red}{-1}, \color{green}{+1}, \color{red}{-1}), (\color{red}{-1}, \color{green}{+1}, \color{green}{+1}), (\color{green}{+1}, \color{red}{-1}, \color{red}{-1}), (\color{green}{+1}, \color{red}{-1}, \color{green}{+1}), (\color{green}{+1}, \color{green}{+1}, \color{red}{-1}), (\color{green}{+1}, \color{green}{+1}, \color{green}{+1})\},$$
so we have
\begin{align*}
C_3 = & \: (\color{red}{-1})(\color{red}{-1})(\color{red}{-1})  \cdot f\Big((\color{red}{-1})x_1 + (\color{red}{-1})x_2 + (\color{red}{-1})x_3\Big) \:+ \\
& \: (\color{red}{-1})(\color{red}{-1})(\color{green}{+1})  \cdot f\Big((\color{red}{-1})x_1 + (\color{red}{-1})x_2 + (\color{green}{+1})x_3\Big) \:+ \\
& \: (\color{red}{-1})(\color{green}{+1})(\color{red}{-1})  \cdot f\Big((\color{red}{-1})x_1 + (\color{green}{+1})x_2 + (\color{red}{-1})x_3\Big) \:+ \\
& \: (\color{red}{-1})(\color{green}{+1})(\color{green}{+1})  \cdot f\Big((\color{red}{-1})x_1 + (\color{green}{+1})x_2 + (\color{green}{+1})x_3\Big) \:+ \\
& \: (\color{green}{+1})(\color{red}{-1})(\color{red}{-1})  \cdot f\Big((\color{green}{+1})x_1 + (\color{red}{-1})x_2 + (\color{red}{-1})x_3\Big) \:+ \\
& \: (\color{green}{+1})(\color{red}{-1})(\color{green}{+1})  \cdot f\Big((\color{green}{+1})x_1 + (\color{red}{-1})x_2 + (\color{green}{+1})x_3\Big) \:+ \\
& \: (\color{green}{+1})(\color{green}{+1})(\color{red}{-1})  \cdot f\Big((\color{green}{+1})x_1 + (\color{green}{+1})x_2 + (\color{red}{-1})x_3\Big) \:+ \\
& \: (\color{green}{+1})(\color{green}{+1})(\color{green}{+1})  \cdot f\Big((\color{green}{+1})x_1 + (\color{green}{+1})x_2 + (\color{green}{+1})x_3\Big).
\end{align*}
A: 
Using $[n]:=\{1,2,\ldots,n\}$ we can write $C_n, n\geq 1$ as
\begin{align*}
\color{blue}{C_n=\sum_{S\subseteq [n]}(-1)^{\left|[n]\setminus S\right|}f\left(\sum_{j\in S}x_j-\sum_{j\in[n]\setminus S}x_j\right)}\tag{1}
\end{align*}

Example: $n=3$: Following (1) we can write $C_3$ as
\begin{align*}
C_3&=(-1)^{\left|[3]\setminus[3]\right|}f(x_1+x_2+x_3)\\
&\qquad+(-1)^{\left|[3]\setminus\{1,2\}\right|}f(x_1+x_2-x_3)
+(-1)^{\left|[3]\setminus\{1,3\}\right|}f(x_1+x_3-x_2)\\
&\qquad+(-1)^{\left|[3]\setminus\{2,3\}\right|}f(x_2+x_3-x_1)\\
&\qquad+(-1)^{\left|[3]\setminus\{1\}\right|}f(x_1-x_2-x_3)
+(-1)^{\left|[3]\setminus\{2\}\right|}f(x_2-x_1-x_3)\\
&\qquad+(-1)^{\left|[3]\setminus\{3\}\right|}f(x_3-x_1-x_2)\\
&\qquad+(-1)^{\left|[3]\setminus\emptyset\right|}f(-x_1-x_2-x_3)
\end{align*}
in accordance with OP's stated $C_3$.
A: Based on @VTand's answer I think you can also write
$$C_n = \sum_{t_1, t_2, \dots, t_n = 0}^1  \epsilon^{t_1}\epsilon^{t_2}\dots \epsilon^{t_n} ~f\Big(\epsilon^{t_1}x_1 + \epsilon^{t_1} x_2 + \dots + \epsilon^{t_1} x_n\Big), $$ where $\epsilon = -1$. In relation to @VTand's example, we have:
\begin{align*}
C_3 = & \: (\color{red}{-1})^{1}(\color{red}{-1})^{1}(\color{red}{-1})^{\color{blue}{1}}  \cdot f\Big((\color{red}{-1})^{\color{blue}{1}}x_1 + (\color{red}{-1})^{\color{blue}{1}}x_2 + (\color{red}{-1})^{\color{blue}{1}}x_3\Big) \:+ \\
& \: (\color{red}{-1})^{\color{blue}{1}}(\color{red}{-1})^{\color{blue}{1}}(\color{green}{-1})^{\color{blue}{0}}  \cdot f\Big((\color{red}{-1})^{\color{blue}{1}}x_1 + (\color{red}{-1})^{\color{blue}{1}}x_2 + (\color{green}{-1})^{\color{blue}{0}}x_3\Big) \:+ \\
& \: (\color{red}{-1})^{\color{blue}{1}}(\color{green}{-1})^{\color{blue}{0}}(\color{red}{-1})^{\color{blue}{1}}  \cdot f\Big((\color{red}{-1})^{\color{blue}{1}}x_1 + (\color{green}{-1})^{\color{blue}{0}}x_2 + (\color{red}{-1})^{\color{blue}{1}}x_3\Big) \:+ \\
& \: (\color{red}{-1})^{\color{blue}{1}}(\color{green}{-1})^{\color{blue}{0}}(\color{green}{-1})^{\color{blue}{0}}  \cdot f\Big((\color{red}{-1})^{\color{blue}{1}}x_1 + (\color{green}{-1})^{\color{blue}{0}}x_2 + (\color{green}{-1})^{\color{blue}{0}}x_3\Big) \:+ \\
& \: (\color{green}{-1})^{\color{blue}{0}}(\color{red}{-1}^{\color{blue}{1}})(\color{red}{-1})^{\color{blue}{1}}  \cdot f\Big((\color{green}{-1})^{\color{blue}{0}}x_1 + (\color{red}{-1})^{\color{blue}{1}}x_2 + (\color{red}{-1})^{\color{blue}{1}}x_3\Big) \:+ \\
& \: (\color{green}{-1})^{\color{blue}{0}}(\color{red}{-1})^{\color{blue}{1}}(\color{green}{-1})^{\color{blue}{0}}  \cdot f\Big((\color{green}{-1})^{\color{blue}{0}}x_1 + (\color{red}{-1})^{\color{blue}{1}}x_2 + (\color{green}{-1})^{\color{blue}{0}}x_3\Big) \:+ \\
& \: (\color{green}{-1})^{\color{blue}{0}}(\color{green}{-1})^{\color{blue}{0}}(\color{red}{-1})^{\color{blue}{1}}  \cdot f\Big((\color{green}{-1})^{\color{blue}{0}}x_1 + (\color{green}{-1})^{\color{blue}{0}}x_2 + (\color{red}{-1})^{\color{blue}{1}}x_3\Big) \:+ \\
& \: (\color{green}{-1})^{\color{blue}{0}}(\color{green}{-1})^{\color{blue}{0}}(\color{green}{-1})^{\color{blue}{0}}  \cdot f\Big((\color{green}{-1})^{\color{blue}{0}}x_1 + (\color{green}{-1})^{\color{blue}{0}}x_2 + (\color{green}{-1})^{\color{blue}{0}}x_3\Big).
\end{align*}
