[sum_kdbinom{n}{k}^2dbinom{3n+k}{2n}=dbinom{3n}{n}^2
]
[egin{aligned}
&=sum_kinom nk^2sum_tinom {3n}{2n-t}inom kt\
&=sum_tinom {3n}{2n-t}sum_kinom nk^2inom kt\
&=sum_tinom {3n}{2n-t}inom ntsum_kinom nkinom {n-t}{k-t}\
&=sum_tinom {3n}{2n-t}inom ntsum_kinom nkinom {n-t}{n-k}\
&=sum_tinom {3n}{2n-t}inom ntinom{2n-t}{n}\
&=sum_tinom {3n}{n}inom ntinom{2n}{n-t}\
&=inom {3n}nsum_tinom ntinom{2n}{n-t}\
&=inom {3n}{n}^2
end{aligned}
]
完 全 胜 利