The question title pretty much says it all.
In derivations where there's some fancy footwork being done with indices, I find expressions like
$$\sum_{i \neq k} x_i$$
a bit too vague. On the other hand, this is explicit enough, but just too unwieldy and awkward-looking:
$$\sum_{i \in \{0, 1, \dots,k-1,k+1,\dots,n-1,n\}} x_i$$
(Of course, I can invent my own notation, but I consider this the "the nuclear option.")
Answer
I've seen $\{0,\dots,\hat{k},\dots,n\}$, but that's not much shorter.
Alternatively, $[n]$ is often shorthand for $\{1,\dots, n\}$, so you could use $[n]\setminus\{k\}$.
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