The Roman factorial is an extension of the ordinary factorial into the negative integers.[1] It is defined as
\begin{eqnarray*} \lfloor n\rceil! &=& n! &\text{for }n \geq 0,\\ \lfloor n\rceil! &=& \displaystyle\frac{(-1)^{-n - 1}}{(-n - 1)!}&\text{for }n < 0.\\ \end{eqnarray*}
It satisfies the identity \(\lfloor n\rceil! = \lfloor n\rceil \lfloor n - 1\rceil!\), where \(\lfloor 0\rceil = 1\) and \(\lfloor n\rceil = n\) for all other \(n\).
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Main article: Factorial
Multifactorials: Double factorial · MultifactorialFalling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial