Prove that C(0)^(2)-C(1)^(2)+C(2)^(2)-...

Prove that
`C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+….+(-1)^(n).C_(n)^(2)={{:((-1)^(n//2)""^(n)C_(n//2)",","if n is even"),(" "0" ,","if n is odd"):}`

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