As many mathematicians known

there exists some kind of relationship between convolution of the central binomial coeffcients n k=0 2k k 2(n k) n k and the cardinality 4 n of some combinatorial structures which can be expressed in the following formula 4n = n k=0 2k k 2(n k) n k.With respect to the formula

We first give

a combinatorial interpretation to n k=0 2k k 2(n k) n k and 4 n using binary sequences

and then construct an algorithm to prove the identity by establishing a correspondence between them.