组合恒等式证明——「Zeilberger 老爷子的 T 恤上写了啥？」
\[\sum_k\dbinom{n}{k}^2\dbinom{3n+k}{2n}=\dbinom{3n}{n}^2
\]
\[\begin{aligned}
&=\sum_k\binom nk^2\sum_t\binom {3n}{2n-t}\binom kt\\
&=\sum_t\binom {3n}{2n-t}\sum_k\binom nk^2\binom kt\\
&=\sum_t\binom {3n}{2n-t}\binom nt\sum_k\binom nk\binom {n-t}{k-t}\\
&=\sum_t\binom {3n}{2n-t}\binom nt\sum_k\binom nk\binom {n-t}{n-k}\\
&=\sum_t\binom {3n}{2n-t}\binom nt\binom{2n-t}{n}\\
&=\sum_t\binom {3n}{n}\binom nt\binom{2n}{n-t}\\
&=\binom {3n}n\sum_t\binom nt\binom{2n}{n-t}\\
&=\binom {3n}{n}^2
\end{aligned}
\]
完 全 胜 利
