I'm a math teacher now, although a few years ago I was finishing up my M.Ed. As part of my studies, I was tasked with conducting my own study of high school level topics and finding unique results. As a student, I had always been fascinated with the idea of converting sequences in explicit form into recursive form, especially since nowhere could I find a general formula.
The images below are my submission to my graduate school and an explanation as to the formula I arrived at for converting polynomial sequences in explicit form into their recursive counterparts.
My BIG question is why? Why does this formula involve the nth derivative, the nth factorial, and an alternating sign?
For an analytic function such as a polynomial, we have Taylor series:
$f(t)=\sum\limits_{n=0}^\infty \dfrac{f^{(n)}(x)}{n!}(t-x)^n$.
Take $t=x-1$ to get your formula.