Why does this process generate the factorial of the exponent? Consider the process of taking a series of numbers and constructing a new series consisting of the difference between consecutive terms, and repeating this until a constant is reached:
$$2,8,18,32,50\\6,10,14,18\\4,4,4$$
When this process is applied to sequences of the form $f(n) = n^a$, the constant reached seems to always be $a!$:
$$1,2,3\\1,1$$
$$1,4,9,16\\3,5,7\\2,2\\$$
$$1,8,27,64,125\\7,19,37,61\\12,18,24\\6,6$$
$$1,16,81,256,625,1296\\15,65,175,369,671\\50,110,194,302\\60,84,108\\24,24$$
Can it be proven?

Yes, it always yields the factorial.
The way you describe to construct each new sub-series from the one above it is similar to taking the derivative of a power function, but at discrete intervals. The rule for taking the derivative of a power function is that $\frac{d}{dx}x^a=ax^{a-1}$. Repeatedly taking this derivative until there is a constant (the power is 0) means that the final coefficient in $cx^0$ will be $a(a-1)(a-2)\dots(2)(1)$, or $a!$.
