The Roman factorial is an extension of the ordinary factorial into the negative integers. It is defined as \begin{eqnarray*} \lfloor nceil! &=& n! & ext{for }n \geq 0,\\ \lfloor nceil! &=& \displaystyle\frac{(-1)^{n - 1}}{(-n - 1)!}& ext{for }n < 0.\\ \end{eqnarray*} It satisfies the identity \(\lfloor nceil! = \lfloor nceil \lfloor n - 1ceil!\), where \(\lfloor 0ceil = 1\) and \(\lfloor nceil = n\) for all other \(n\).