The tetrofactorial is a function based on tetration and factorial.[1] It's defined as \(n!2 = n↑↑(n-1)↑↑(n-2)↑↑...↑↑2↑↑1\) using hyperfactorial array notation. This has the growth rate of almost pentation.

Below are the first few tetrofactorials:

\(1!2 = 1\)

\(2!2 = 2↑↑1 = 2\)

\(3!2 = 3↑↑2↑↑1 = 3↑↑2 = 27\)

\(4!2 = 4↑↑3↑↑2↑↑1 = 4↑↑3↑↑2 = 4↑↑27\) ~ E153#25

\(5!2 = 5↑↑4↑↑3↑↑2↑↑1 = 5↑↑4↑↑3↑↑2 = 5↑↑4↑↑27\)

Sources

See also

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 · 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
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