Curious if this function is called anything.

[hddd123456789]

Hey everyone, I was just playing around with factorial expansion and after grouping some terms I noticed a pattern for which I made a function. I am happy to provide more details as needed, but for now I have attached an image of the function and am curious, as per the title, whether this is known and if so if it has a name and/or interesting properties. I forgot to put this on the image but I was only considering positive naturals for a and n when defining the function.

Thanks!

 

[hddd123456789]

Sorry, I should've posted this after working out the kinks. The a-n-1, a-n-2 part should be a-(n-1), a-(n-2), ... and k_n=n-1 should be k_n=n.

 

[Pythagorean]

I don't know, but I think it could be simplified with pi notation:

http://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation

 

[hddd123456789]

Actually, I figured out that these are (unsigned) Stirling numbers of the first kind (save where n=0), with 'a' being the rows of its pyramid, and 'n' being the column in each row. Ironic that I called the function S. I attached a simpler (and more accurate) version for anyone interested.

 

[CellCoree]

Hello there! It's great to see you exploring factorial expansion and noticing patterns in your calculations. It sounds like you have found a function that may have some interesting properties. Without seeing the image or more details, it's difficult to say whether this function has a specific name or if it is a known function. However, I would suggest doing some research to see if any existing functions or theorems relate to your discovery. If not, you could consider naming the function yourself and possibly even publishing your findings if they are significant. Keep exploring and don't be afraid to reach out to other mathematicians for their input and insights. Happy calculating!