A combinatorial proof of two equivalent identities by free 2-Motzkin paths (Q2855616)

In 1902, MacMahon derived a formula for the sum of cubes of binomial coefficients. \textit{H. W. Gould} [Ars Comb. 86, 161--173 (2008; Zbl 1221.33017)] derived another identity from Carlitz's formula. Using colored free 2-Motzkin paths, the author gives a combinatorial proof of the resulting identityNEWLINE NEWLINE\[NEWLINE \sum\limits_{0\leq k\leq \frac{n}{2}} \binom{n}{2k}\binom{2k}{k}\binom{n+k}{k}2^{n-2k} =\sum\limits_{0\leq k\leq \frac{n}{2}} \binom{n}{2k}\binom{2k}{k}\binom{2n-2k}{n-k}. NEWLINE\]