User blog:Undeadlift/Ultra-Factorial Functional Array

The Ultra-Factorial Functional Array (UFA) is a method of creating very large numbers using factorials and recursion in a simple linear notation.

Definitions and Examples

 * \(|a,b| = a!^b\), where \(b\) is the number of \(!\)'s.


 * \(|1,0| = 1\)
 * \(|1,1| = 1! = 1\)
 * \(|2,1| = 2! = 2\)
 * \(|3,1| = 3! = 6\)
 * \(|2,2| = 2!! = 2\)
 * \(|3,2| = 3!! = 6! = 720\)
 * \(|4,2| = 4!! = 24! = 620,448,401,733,239,439,360,000\)
 * \(|3,3| = 3!!! = 6!! = 720!\)

Extended Notation
In extended notation, another input is added into the array in order to operate a system of recursion.
 * \(|a,b,0| = |a,b|\)
 * \(|a,b,c| = |a,|a,b|,c-1\)


 * \(|3,1,0| = |3,1| = 3! = 6\)
 * \(|3,1,1| = |3,|3,1|,0| = |3,6| = 3!!!!!!\)
 * \(|3,2,1| = |3,|3,2|,0| = |3,720| = 3!!!...!\), where there are 720 number of \(!\)'s
 * \(|3,3,1| = |3,|3,3|,0| = |3,720!| = 3!!!...!\), where there are 720! number of \(!\)'s
 * \(|3,3,2| = |3,|3,3|,1| = |3,720!,1| = |3,|3,720!|,0| = |3,|3,720!|| = 3!!!...!\), where there are |3,720!| number of \(!\)'s

Further Extensions
In extended notation, another input is added into the array in order to operate a system of recursion.
 * \(|a,b,c,0| = |a,b,c|\)
 * \(|a,b,c,d| = |a,b,|a,b,c|,d-1\)
 * \(|a,b,c,d,0| = |a,b,c,d|\)
 * \(|a,b,c,d,e| = |a,b,c,|a,b,c,d|,e-1\)
 * \(|a,b,c,...x,y,z| = |a,b,c,...x,|a,b,c,...x,y|,z-1|\)


 * \(|3,3,3,0| = |3,3,3| = |3,|3,3|,2| = |3,720!,2| = |3,|3,720!|,1|\)...
 * \(|3,1,1,1| = |3,1,|3,1,1|,0| = |3,1,|3,|3,1||| = |3,1,|3,6|| = |3,1,3!!!!!!| = |3,|3,1|,3!!!!!-1| = |3,6,3!!!!!!-1| = |3,|3,6|,3!!!!!!-2| = |3,|3,3!!!!!!|,3!!!!!!-2| = |3,3!!!...!,3!!!!!!-2|\)...