| Nested factorial notation | |
|---|---|
| Notation | \(n!^m\) |
| Type | Combinatorial |
| Based on | Factorial |
| Growth rate | \(f_{3}(n)\) |
| Author | Aarex Tiaokhiao |
Nested factorial notation is an extension of the factorial function, created by Aarex Tiaokhiao.[1] It is formally defined as follows:
n!m = ((...(((n!)!)!...)!)! w/m nested factorials (not to be confused with n! to the power of m).
For example, 3!3 = ((3!)!)! = (6!)! = 720! It's decimal expansion is:
Extended Nested Factorial Notation[]
| Extended Nested Factorial Notation | |
|---|---|
| Notation | \(n!^m[x]\) |
| Type | Combinatorial |
| Based on | Nested factorial notation |
| Growth rate | \(f_{\omega}(n)\) |
| Author | Aarex Tiaokhiao |
Aarex defined n![2], which is equal to n!n. Also he defined multorial of n, is same at n![2]. This can also be written as n% using warp notation.
Next is n!2[2], is equal to (n!n)!n!n, or (n![2])![2].
Then n!3[2] = (n!n)!2[2] = ((n![2])![2])![2], and so on.
He defined n!n[2], is same at n![3]. And also defined n![4], is equal to n!n[3].
In general, n![m] is equal to n!n[m-1]. For example, 7![3], is equal to 7!7[2].
The limit of this notation is n![n].
Multi-entry Factorial Notation[]
| Multi-entry Factorial Notation | |
|---|---|
| Notation | \(n!^m[a,b,c,...]\) |
| Type | Combinatorial |
| Based on | Nested factorial notation |
| Growth rate | \(f_{\omega^\omega}(n)\) |
| Author | Aarex Tiaokhiao |
Aarex defined n![1,2], which is equal to n![n].
Then n![2,2] = n!n[1,2], n![3,2] = n!n[2,2], etc.
The same pattern continues for more entries, each corresponding to the next exponent on \(\omega\) for \(f_{\omega^m}(n)\) in the FGH.
Sources[]
See also[]
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