Messy sum simplifies to a binomial coefficient!

Probability Level 5

Let S(n)=i=0n2i(ni)(nini2). 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)=(M2017)S(2017) = \dbinom{M}{2017} , find MM.


Notations:

  • \lfloor \cdot \rfloor denotes the floor function.
  • (MN)=M!N!(MN)! \dbinom MN = \dfrac {M!}{N! (M-N)!} denotes the binomial coefficient.

Inspiration.

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