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**.

We could then simply notate it as this:

## Definition

Define as:

Also, let be equal to multiplication.

Then,