Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence

Theorem
Let $\sequence a, \sequence b$ be real sequences which satisfy the condition:


 * $a_n = \displaystyle \sum_{r \mathop = 0}^n \binom n r b_r$

Then:
 * $\displaystyle \paren {-1}^n b_n = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s a_s$

Proof
For $n - r > 1$ we can use the Binomial Theorem with $x = 1$ and $y = -1$:


 * $0 = \paren {1 - 1}^{n - r} = \displaystyle \sum_{s \mathop = 0}^{n \mathop - r} \binom {n - r} s \paren {-1}^s$

Hence all terms of $(1)$ vanish except for where $n - r$.

That term is:
 * $\paren {-1}^n b_n$