For questions related to left factorials, $!n$ (that is, either subfactorials or factorial sums).
Left factorials, $!n$, may refer to either subfactorials or to factorial sums.
The $n$th subfactorial (also called the derangement number) is the number of permutations of $n$ objects in which no object appears in its natural place (i.e., "derangements").
The first few values of $!n$ for $n=1, 2,\dots$ are $0, 1, 2, 9, 44, 265, 1854, 14833,\dots$ (OEIS A$000166$).
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As factorial sums, $!n=\sum^{n-1}_{k=0}k!$
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