Ways to computationally simplify a combinatorial expression? I have the following expression,
$$\sum_{m=1}^{n}\sum_{k=m}^{n} (-1)^{k-m} {k\choose m} P_k$$
where $P_k$ is an arbitrary function only depending on $k$.
Now it is easy to see that this expression will compute the same things over and over again. Is there a more compact representation for this which overcomes this problem? (I don't want a computational solution but rather an algebraic solution which reflects the savings in computation)

Peter Foreman in the comments is correct. Reverse the order of summation and use the binomial formula:
$$\begin{align*}
\sum_{m=1}^n\sum_{k=m}^n(-1)^{k-m}\binom{k}mP_k&=\sum_{k=1}^nP_k\sum_{m=1}^k(-1)^{k-m}\binom{k}m\\
&=\sum_{k=1}^nP_k\left(\underbrace{\sum_{m=0}^k(-1)^{k-m}\binom{k}m}_{=0}-(-1)^k\right)\\
&=\sum_{k=1}^n(-1)^{k+1}P_k
\end{align*}$$
