Let n be a positive integer. Prove that

\(\displaystyle \sum_{k=0}^n (-1)^k \binom{2n+2k}{n+k} \binom{n+k}{2k}=(-4)^n\).

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-14.

Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 정진명 (수리과학과 2007학번, +3), 박민재 (KSA-한국과학영재학교, +3), 오성진 (Princeton Univ.), Abhishek Verma (GET-SKEC NDEC, New Delhi).

Here are some interesting solutions.

• Solution by 오성진 using the Cauchy integral formula.
• Solution by 정진명 NOT using the binomial theorem.
• Two solutions by 박민재 : (1) A combinatorial solution, (2) Using generating functions and Lagrange’s Inversion Theorem.