Torian

The Torian is a function invented by Aalbert Torsius. 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 .
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 ==
Category:Functions
Category:Factorials
Category:Notations