Messy sum simplifies to a binomial coefficient!
Discrete Mathematics
Level
5
Let \[ S(n)=\sum_{i=0}^n 2^i{n\choose i}{n-i\choose \left\lfloor\frac{n-i}{2}\right\rfloor}. \]
If \(S(2017) = \dbinom{M}{2017} \), find \(M\).
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Notations:
- \( \lfloor \cdot \rfloor \) denotes the floor function.
- \( \dbinom MN = \dfrac {M!}{N! (M-N)!}\) denotes the binomial coefficient.
by
Jessica Wang