In terms of hypergeometric series, the sum is $_3F_2(-n, 1+n, 1/2;1,3/2;1)$ and the identity is a special case of [Saalsch\"utz's theorem][1] (also called the Pfaff-Saalsch\"ute theorem), one of the standard hypergeometric series identities. 


  [1]: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function#Saalsch%C3%BCtz's_theorem