Let $i$, $k$ be integers such that $2 \leq i \leq k$. I would like to show that the sum
$$
\sum\_{j=1}^{i-1} \frac{(-1)^{j-1}(i-j)^k}{(i-j)! (j-1)!}
$$
is positive. I have carried out extensive numerical experiments to check this for small values of $k$. In fact, much more should be true. Define polynomials
$$
U(x)=(x+i-1)^k
$$
and
$$
V(x)=x(x+1)\cdots(x+i-1).
$$
Let $Q$ and $R$ be the quotient and remainder on dividing $U$ by $V$. The above sum is the leading coefficient of $R$. It seems that all the coefficients of $Q$ and $R$ are always positive, and it would be nice to prove this, but I only need the positivity of the above sum. This question has applications for proving the irrationality of certain series.



Positivity of a finite sum
These are [Stirling numbers of the second kind](http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind). More precisely
your sum is S(k,i-1) where $S$ denotes Stirling number of the second kind.


