Editing Derangement (section)

=== Asymptotic expansion in terms of Bell numbers ===
An asymptotic expansion for the number of [[derangements]] in terms of [[Bell numbers]] is as follows:
<math display="block">!n = \frac{n!}{e} + \sum_{k=1}^m \left(-1\right)^{n+k-1}\frac{B_k}{n^k} + O\left(\frac{1}{n^{m+1}}\right),</math>
where <math>m</math> is any fixed positive integer, and <math>B_k</math> denotes the <math>k</math>-th [[Bell number]]. Moreover, the constant implied by the [[big O notation|big O]]-term does not exceed <math>B_{m+1}</math>.<ref>{{cite journal
 | last = Hassani  | first = Mehdi
 | year = 2020
 | title = Derangements and Alternating Sum of Permutations by Integration
 | journal = [[Journal of Integer Sequences]]
 | volume = 23 | at = Article&nbsp;20.7.8  | no-pp=yes
 | url = https://cs.uwaterloo.ca/journals/JIS/VOL23/Hassani/hassani5.html
}}</ref>