Solution to differences of differences...

In the post differences of differences, I make an observation that
if you keep taking the differences of consecutive square numbers,
cube numbers and fourth powers, you will eventually get a line of
numbers which are exactly the same. I had said that the repeating
number would be the exponent's factorial and that was the case up
to 5. I didn't know why this happened, but I think I've found out:
If you write out the line of numbers algebraically and do the
subtraction, then you get a clear (oh really?) answer.

SIDE NOTE: My maths teacher is really insistent on me writing
complete answers with statements, so here goes:

So, the first line is No, with statements. 

Let the exponent = k
Therefore, 
1st line - 1^k, 2^k, 3^k, 4^k, 5^k, 6^k, 7^k and so on...

2nd line - 2^k-1^k, 3^k-2^k, 4^k-3^k, 5^k-4^k and so on...

3rd line - 3^k-2(2^k)+1^k, 4^k-2(3^k)+2^k, 5^k-2(4^k)+3^k and so on...
.
.
.
kth line - immensely complex maths-stuff

After working for 30 minutes on this, I felt that the effort
required to create a solution was not worth it. Therefore,
I leave it as an open problem.