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I found the following formula in a book without any proof:

$$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$

This does not seem to follow immediately from the basic binomial identities. I would like to know how to prove this, and any relevant references.

Remark : This question has been asked previously on math.SE without receiving any complete answers.

I found the following formula in a book without any proof:

$$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$

This does not seem to follow immediately from the basic binomial identities. I would like to know how to prove this, and any relevant references.

Remark : This question has been asked previously on math.SE without receiving any complete answers.