What I am talking about is a fact, that if we write down n-th powers of consecutive natural numbers in a row, and then on the next row between each two numbers write their difference and repeat this algorithm, then on the n-th row we always get a row of $n!$'s. So why it is like that? The only way I can associate power with factorial is series, but I think it's not the case. Here is an example of the thing I am talking about:
1 32 243 1024 3125 7776 31 211 781 2101 4651 180 570 1320 2550 390 750 1230 360 480 120
n-th (last) row (if we assume that first one is 0th) is always n! (in this case 5!).
I hope my question is clear. Thanks in advance!
Cheers