Differences of differences ...

I was reading my copy of  THE PERFECT SAUSAGE

when I came across something interesting : 

A section displaying what happens when you mess around with sequences of 

numbers. If you write all the square numbers in a line, then the each difference between 

consecutive terms will be the odd numbers in a line. The difference between 

consecutive terms will now be a straight line of 2's, since every odd number is separated 

by a difference of 2.

    This simplifying down to a contiguous line caught my attention. I experimented with 

cube numbers, taking the repeated differences of terms. Here it converged down to a

single line of 6's.

     One might think it should have been a line of 3's such that the final difference should 

match the exponent : Powers of 2 come to 2, so powers of 3 come to 3.

      Experimenting further, I wrote out a line of power of 4's : 1, 16, 27, 64 ...

The differences of the differences of the differences of the differences 

fiiinaaallyyy come to : a straight line of 24's. [Disappointingly small answer for so much toil]

      You may have noticed a pattern at this point, I definitely did :

2, 6, 24 ... are simply the factorials of the exponent :

2! = 2 ,  3! = 6 ,  4! = 24  ,   .... and so on.

With powers of 5's, I wont bore you with the intermediate steps, I'll steam right through :

A STRAIGHT LINE OF 120's. And as predicted, 5! = 120.

I still do not have a proof.

If someone has struck on a proof which I oversaw, please enter it in the comments section.