While exploring the Baxter sequences from [my earlier MO post][1], I obtained a rather curious identity (not listed on OEIS either). I usually try to employ the Wilf-Zeilberger (WZ) algorithm to justify such claims. To my surprise, WZ offers **two different** recurrences for each side of this identity.

So, I would like to ask:

>**QUESTION.** Is there a conceptual or **combinatorial reason** for the below equality?
$$\frac1n\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}
=\frac2{n+2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n-1}k\binom{n+2}{k+2}.$$

**Remark.** Of course, one gets an alternative formulation for the Baxer sequences themselves:
$$\sum_{k=0}^{n-1}\frac{\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}}{\binom{n+1}1\binom{n+1}2}
=\sum_{k=0}^{n-1}\frac{\binom{n+1}k\binom{n-1}k\binom{n+2}{k+2}}{\binom{n+1}1\binom{n+2}2}.$$

[1]: https://mathoverflow.net/questions/412324/closed-formula-for-1-baxter-sequences