Sum over k of m Choose k by k minus m over 2

Theorem

$\ds \sum_{k \mathop = 0}^n \binom m k \paren {k - \dfrac m 2} = -\dfrac m 2 \binom {m - 1} n$

where $\dbinom m k$ etc. are binomial coefficients.

$\ds \sum_{k \mathop = 0}^n \binom m k \paren {k - \dfrac m 2}$

$=$

$\ds \sum_{k \mathop = 0}^n k \binom m k - \dfrac m 2 \sum_{k \mathop = 0}^n \binom m k$

$\ds $

$=$

$\ds \sum_{k \mathop = 0}^n m \binom {m - 1} {k - 1} - \dfrac m 2 \sum_{k \mathop = 0}^n \binom m k$

$\ds $

$=$

$\ds \dfrac m 2 \sum_{k \mathop = 0}^n \paren {2 \binom {m - 1} {k - 1} - \binom m k}$

$\ds $

$=$

$\ds \dfrac m 2 \sum_{k \mathop = 0}^n \paren {2 \binom {m - 1} {k - 1} - \paren {\binom {m - 1} {k - 1} + \binom {m - 1} k} }$

$\ds $

$=$

$\ds \dfrac m 2 \sum_{k \mathop = 0}^n \paren {\binom {m - 1} {k - 1} - \binom {m - 1} k}$

simplifying

$\ds $

$=$

$\ds \dfrac m 2 \paren {\binom {m - 1} {-1} - \binom {m - 1} n}$

as the series telescopes

$\ds $

$=$

$\ds -\dfrac m 2 \binom {m - 1} n$

Definition of Binomial Coefficient for negative lower index

$\blacksquare$