Let $k,p$ be positive integers. Is there a closed form for the sums 


$$\sum\_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$


$$\sum\_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$


(where 'closed form' should be interpreted as usual, i.e. meaning free of sums and hypergeometric functions).


We know that the first sum has generating function $(1+z)^k/(1-z)^{k+1}$, and the second sum has generating function $(1+z)^{k-1}/(1-z)^{k+1}$, but that doesn't help me find a closed form so far. 



Yet another sum involving binomial coefficients
Your first sum is the Delannoy number $D(k,p)$. See [OEIS sequence A008288](http://oeis.org/A008288)


