Is it an Algebra Problem or Calculus?

Algebra Level 5

\[\Large{ \sum_{i=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \binom{n-i-1}{i} \dfrac{2^{n-2i-1}}{n-2i} = \ ? }\]

Evaluate the above summation for \(n=20\).

Note - You have to use a calculator at the last step of the solution if you generalize it for all \(n\).

Bonus - Generalize it for all \(n \in \mathbb Z^+ \).

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