User blog:King2218/Extended Factorial Notation

Hello!

This is my first post so I apologize for the growth of the function and the informal definition. (I'm also not sure if someone else has done this)

So, basically, I was challenging myself in getting the factorials of numbers from 1 to 30. At some point, I got bored and decided to extend the notation.

The first one is the level-2 factorial: ​I ran into a lot of trouble defining that one but anyway here are some examples: ​For the level-3 factorial it would need a bit of examining to understand: ​You might need an example: ​3!!! might be small but 4!!! is HUGE.
 * n!! = (n(n-1) (n-2) ... ) * ((n-1)(n-2) (n-3) ...  ) * ((n-2)(n-3) (n-4) ...  ) * ...
 * 3!! = 32 1 * 21 * 1 = 9*2 = 18
 * 4!! = 43 2 1 * 32 1 * 21 * 1 = 4718592
 * n!!! = ((n^^(n-1)^^(n-2)...)((n-1)^^(n-2)^^.....) .... )*(((n-1)^^(n-2)^^(n-3)^^...)((n-2)^^(n-3)^^...) ... )*...
 * 3!!! = ((3^^2^^1)^(2^^1)^1)*((2^^1)^1)*1 = 729*2*1 = 1458

We could then simply notate it as this: The reason for the parentheses bounding the m is because I thought that I could use some super-dimensional levels and the like for m.
 * n!!!!...m !'s...!!! = n!(m)

For example, m could be (1, 2), (1, 1, 1, 2) or even (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 4!!! 1's ..., 1, 2).

Anyway that's it!