Is the following identity known?


$$\sum\limits\_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n+k}{n-k}\binom{2k}{k}=
\frac{1}{2n+1}$$


I have not found it in the following book:


* Henry Wadsworth Gould, [Combinatorial identities: a standardized set of tables listing 500 binomial coefficient summations](https://books.google.com/books/about/Combinatorial_identities.html?id=idI-AAAAIAAJ), Morgantown, West Virginia, 1972.



New binomial coefficient identity?
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 [Saalschütz's theorem](https://en.wikipedia.org/wiki/Generalized_hypergeometric_function#Saalsch%C3%BCtz's_theorem) (also called the Pfaff-Saalschütz theorem), one of the standard hypergeometric series identities. 


A more general identity, also a special case of Saalschütz's theorem, is 
$$\sum\_{k=0}^n (-1)^k\frac{a}{a+k}\binom{n+k+b}{n-k}\binom{2k+b}{k}
 = \binom{n+b-a}{n}\biggm/\binom{n+a}{n}.$$
The O.P.'s identity is the case $a=1/2, b=0$.


