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Somewhere I saw the notation $\{a_{k}\}^{n}_{0}$ used. I believe it was used for a summation, but I'm not sure. What does this actually mean and can also use it to describe the set $\{a_0,a_1,...,a_n\}$?

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That is a short way to denote the sequence of elements $$a_k,$$ as $k$ runs over some index set. Here, $k$ runs from $0$ to $n.$

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