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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.

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  • $\begingroup$ Where can I find proof for the second sum in OP's question? $\endgroup$ – Hasek Feb 24 '17 at 10:57
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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.

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