The alternating factorial of a number n is \(\sum^n_{i = 1} (-1)^{n - i} \cdot i!\), or the alternating sum of all the factorials up to n. For example, the alternating factorial of 5 is \(1! - 2! + 3! - 4! + 5!=101\).[1]

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Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial

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