Torian | |
---|---|
Notation | \(T(n)\) |
Type | Combinatorial |
Based on | Factorial |
Growth rate | \(f_{2}(n)\) |
Author | Aalbert Torsius |
The Torian is a function invented by Aalbert Torsius.[1] It is defined as \(T(x) = x!x\), using Torsius' definition of the factorial:
\(x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)\),
\(x!0 = x\)
This definition is a generalization of the ordinary factorial: \(x!1 = x!\).
The first few values of \(x!x\) for \(x = 0, 1, 2, 3, 4, 5, 6, 7, \cdots\) are \(0, 1, 2, 24, 331\,776, 252\,428\,641\,478\,023\,533\,120, 1.8356962141506\cdot 10^{82}, 5.1012625185483\cdot 10^{315}, \cdots\)
Faster method of computation[]
It is possible to calculate T(x) in a much faster way.
\(trn_x(n) = trn_{x-1}(1) + trn_{x-1}(2) + trn_{x-1}(3) \cdots trn_{x-1}(n)\)
\(trn_0(n) = n\)
The process of computing \(trn_x(n)\) can be also shortened using formula \({n^{(x)} \over (n+1)!}\), where \(n^{(x)}\) is the rising factorial.
Here "\(trn_x\)" stands for order-x triangular number.
Then consider how it relates to x-order factorials:
\(n!2 = 2 \times (2 \times 3) \times (2 \times 3 \times 4) \ldots (2 \times 3 \times 4 \ldots (n-1) \times n)\). Multiplication is commutative, and we know that 2 appears in that expression n-1 times, 3 appears n-2 times, and x appears n-(x-1) times.
Thus, \(n!2 = 2^{n-1} \times 3^{n-2} \times 4^{n-3} \times 5^{n-4} \cdots n\) It can be also written as follows: \(n!2 = 2^{trn_0(n-1)} \times 3^{trn_0(n-2)} \times 4^{trn_0(n-3)} \times 5^{trn_0(n-4)} \cdots n\)
By the analogical considerations, \(n!3 = 2^{trn_1(n-1)} \times 3^{trn_1(n-2)} \times 4^{trn_1(n-3)} \times 5^{trn_1(n-4)} \cdots n\)
In general, \(n!x = 2^{trn_{x-2}(n-1)} \times 3^{trn_{x-2}(n-2)} \times 4^{trn_{x-2}(n-3)} \times 5^{trn_{x-2}(n-4)} \cdots n\)
Pseudocode[]
// Torsius' factorial extension function factorialTorsius(z, x): if x = 0: return z result := 1 for i from 1 to z: result := result * factorialTorsius(i, x - 1) return result // Torian function torian(x): return factorialTorsius(x, x)
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