I'm pretty new to this site, so if this is in the wrong section (I think it *is*), please tell me what to do!

First off, I need a better way to word all this...but mostly I'm just looking for an explanation to this:

The difference between integers to the *1st* power is **1**.

$\displaystyle 1^1-0^1=1$

The difference between the difference between integers to the *2nd* power is **2**.

$\displaystyle 2^2-1^2=3$

$\displaystyle 1^2-0^2=1$

$\displaystyle 3-1=2$

The difference between the difference between the difference between integers to the *3rd* power is **6**.

$\displaystyle 3^3-2^3=19$

$\displaystyle 2^3-1^3=7$

$\displaystyle 1^3-0^3=1$

$\displaystyle 19-7=12$

$\displaystyle 7-1=6$

$\displaystyle 12-6=6$

And, the difference between the difference between the difference between the difference between *4th* power is **24**.

$\displaystyle 4^4-3^4=175$

$\displaystyle 3^4-2^4=65$

$\displaystyle 2^4-1^4=15$

$\displaystyle 1^4-0^4=1$

$\displaystyle 175-65=110$

$\displaystyle 65-15=50$

$\displaystyle 15-1=14$

$\displaystyle 110-50=60$

$\displaystyle 50-14=36$

$\displaystyle 60-36=24$

If the exponent for the example above is $\displaystyle n$, then $\displaystyle n!$ is the result (from what I can tell). Why?