As an off-shot of my earlier MO question, I have found a "really cute" identity. The connection is revealed in the limit $q\rightarrow 1$.

So, I would like to ask:

QUESTION.Is there a combinatorial (conceptual) proof of this equality? $$\prod_{i=1}^n\binom{2i}i^2=2^n\prod_{i=1}^n\binom{n+i}i\binom{n}i \qquad \text{or} \qquad \prod_{i=1}^n\binom{2i}i^2=2^n\prod_{i=1}^n\binom{n+i}{i,i,n-i}.$$

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