Yet another sum involving binomial coefficients - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:52:32Z http://mathoverflow.net/feeds/question/70424 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70424/yet-another-sum-involving-binomial-coefficients Yet another sum involving binomial coefficients Manolito Pérez 2011-07-15T11:52:07Z 2011-07-16T00:42:13Z <p>Let $k,p$ be positive integers. Is there a closed form for the sums </p> <p>$$\sum_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$</p> <p>$$\sum_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$</p> <p>(where 'closed form' should be interpreted as usual, i.e. meaning free of sums and hypergeometric functions).</p> <p>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. </p> http://mathoverflow.net/questions/70424/yet-another-sum-involving-binomial-coefficients/70474#70474 Answer by Robert Israel for Yet another sum involving binomial coefficients Robert Israel 2011-07-16T00:42:13Z 2011-07-16T00:42:13Z <p>Your first sum is the Delannoy number $D(k,p)$. See <a href="http://oeis.org/A008288" rel="nofollow">OEIS sequence A008288</a></p>