Consider w(i) = \sum_{j=0}^{i-1} (-1)^j(i-j)^n{i \choose j}
where i=1,2,3,... and n=1,2,3,...
Question: Prove w(i)=0 if i>n.
---------------------------------------------------------------

Furthermore, suppose {f}_{i}(x) = \sum_{j=0}^{i-1} (-1)^j(i-j)^x{i \choose j}
where i=1,2,3,... and x \in \mathbb{R}
Question: Are x=1,2,3,...,i-1 the only zeros of {f}_{i}(x)?

Motivation for w(i) see here: http://www.mathhelpforum.com/math-he...ty-194323.html where it's defined recursively.