If sequences $\{a_k\}$ and $\{b_k\}$ are related by $a_k = \sum_{r=0}^k \binom{k}{r} b_r$, can I compute $b_k$s in $O(1)$ time?
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$