Is there a standard definition for $!n$ like $n!$?
The definition of $n!,n!!,n!!!,\ldots$ is very is easy to find. but I can't find anything about $!n$.
for example, $$5!=1\cdot2\cdot3\cdot4\cdot5\\9!!=1\cdot3\cdot5\cdot7\cdot9$$
but, does it mean? $$!5=\text{?}\\!3=\text{?}\\ \vdots$$ can someone help me, or show me a reference? Thanks in advance.
2026-04-12 16:59:59.1776013199
Factorial sign before a number
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Notation of a preceding exclamation point can represent the subfactorial (a.k.a. derangement number). In this usage, \begin{align*} !0 &= 1 \\ !1 &= 0 \\ !n &= (n-1)(!(n-1) + !(n-2)), n > 1 \text{.} \end{align*}