TITLE: Is there a combinatorial interpretation for this sum?
QUESTION [7 upvotes]: Is there a combinatorial interpretation for:
$$\sum_{i=0}^{n}\binom{3i}{i}\binom{3(n-i)}{n-i}?$$
I do not think there is a simple closed form for it, like: 
$$\sum_{i=0}^{n}\binom{2i}{i}\binom{2(n-i)}{n-i} = 4^{n}.$$
Several types of combinatorial proofs are given for this identity.

REPLY [2 votes]: A good tool to apply for such problems is the Online Encyclopedia of Integer Sequences. I computed the first ten value or so of your function of $n$, searched oeis.org, and found an entry for the sequence that contains formulas, asymptotics, references, and so on.