In the following equation, $$n^s=(n)_s+f(s)$$ What is general form for $f(s)$? Understand that, $$(n)_s=n(n-1)(n-2)\cdots(n-[s-1])=\text{ The Falling Factorial }$$
I have experimented with this equation for $s=\{1,2,3,4\}$. Unless my calculations are horribly mangled, the following table arises: $$ \begin{array}{c|c} s & f(s)\\ \hline 1 & 0\\ 2 & n\\ 3 & 3n^2-2n\\ 4 & 6n^3-11n^2+6n \end{array} $$
I cannot see any reasonable pattern to these values. There isn't an obvious (to me) reccurence relationship, so that method of solving this seems useless. I would appreciate any help on this; it is a personal curiosity of mine. I'm a freshman in highschool. So, I would appreciate elaboration on any complex or complicated methods.
To be more specific, I would like $f(s)$ defined in the form of a polynomial. This is the particular form I was considering: $$f(s)=c_{s-1}n^{s-1}+c_{s-2}n^{s-2}+\dots+c_{1}n$$