The following sum has emerged in my research:
$ \sum_{j=k}^{n}\frac{1}{2^j}{j\choose k}{n+k+2\choose n-j}; $
I am looking for a closed form for this sum (without hypergeometric series). I have searched through Graham, Knuth and Patashnik's Concrete Mathematics and could not locate any identities which could be used to simplify this sum, or express it in closed-form (without hypergeometric series).
Does anyone know a way to express this sum in closed-form?