A binomial sum expression - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:55:55Z http://mathoverflow.net/feeds/question/25643 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25643/a-binomial-sum-expression A binomial sum expression unknown (yahoo) 2010-05-23T09:27:50Z 2010-05-23T14:57:24Z <p>Does anyone know how to show the following combinatorial equality, $\sum_{i=0}^{n}\left(n-i\right)^{2}\binom{2n}{i}=n\cdot4^{n-1}$?</p> <p>By the way, this is not a homework problem, otherwise one would be able to search the answer.</p> <p>Thanks.</p> http://mathoverflow.net/questions/25643/a-binomial-sum-expression/25645#25645 Answer by Roland van der Veen for A binomial sum expression Roland van der Veen 2010-05-23T09:36:37Z 2010-05-23T09:36:37Z <p>Try the Wilf-Zeilberger method and its friends. This automatically proves many such (hypergeometric) identities. See the book A = B</p> <p><a href="http://www.math.upenn.edu/~wilf/AeqB.html" rel="nofollow">http://www.math.upenn.edu/~wilf/AeqB.html</a></p> http://mathoverflow.net/questions/25643/a-binomial-sum-expression/25646#25646 Answer by Robin Chapman for A binomial sum expression Robin Chapman 2010-05-23T09:37:02Z 2010-05-23T10:04:34Z <p>It's half the sum of the same thing from $0$ to $2n$, which in turn is easily related to the variance of the number of heads in a sequence of $2n$ tosses of a fair coin.</p> http://mathoverflow.net/questions/25643/a-binomial-sum-expression/25648#25648 Answer by Per Alexandersson for A binomial sum expression Per Alexandersson 2010-05-23T09:38:16Z 2010-05-23T09:38:16Z <p>I suggest you take a look on hypergeometric series.</p>