I have been tinkering with a variation of Pascal's triangle using the differences between the exponents of consecutive numbers to derive a factorial.
Generally, the bottom row starts with 1^n , 2^n , 3^n ... (n+1)^n , (n+2)^n (the bottom row contains n + 2 digits)enter image description here
For example, to calculate 3! (n=3), you start with : 1^3, 2^3, 3^3, 4^3, 5^3
5^3-4^3 = 61, 4^3-3^3 = 37, 3^3-2^3 = 19, 2^3-1^3 = 7
61-37 = 24, 37-19 = 18, 19-7 = 12
24-18 = 6, 18-12 = 6
So the constant is 6, which is 3!
I've tried this with larger numbers and the constant so far has been n!
If this is confusing, I have attempted to attach a page with the hand-written calculations.
Thank you kindly for any constructive help!