Newton explained it all. He made contributions which are super-human to say the least.
Most of the mathematical equations we learnt work due to gravity. (My view.). The sine, cos, e, log. This is how I understand. Because if there is no gravity, then how can a building stand. If a building cannot stand then where is sine.
However, pi is independent of gravity. (Again as I see it).
The n! factorial moves with e function or log function. And is calculated by integral e function and integral log functions. This calculation of n! is due to gravity. You may observe that the calculations due to n/(n+1)! is due to lifting weights. Lifting weights in gravity. Therefore the n! works there.
Now in my earlier post I gave the formula of n!. It is due to pi. And pi is non gravitational. Therefore, it cannot be expected that n! using the formula in my earlier post will strictly match the n! due to e and log and my earlier sigma function which are of gravity.
I hope I have not messed up things. I hope I am able to say my points clearly. And convey easily.
This is how I understand. If you find something wrong here, do convey in blog comments below.
Have a nice day !!