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Prove that (.^(2n)C0)^2-(.^(2n)C1)^2+(.^...

Prove that `(.^(2n)C_0)^2-(.^(2n)C_1)^2+(.^(2n)C_2)^2-..+(.^(2n)C_(2n))^2` = `(-1)^n.^(2n)C_n`.

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`underset(r=0)overset(2n)sum(-1)^(r)(.^(2n)C_(r))^(2)=underset(r=0)overset(2n)sum(-1)^(r).^(2n)C_(r).^(2n)C_(r)`
`= underset(r=0)overset(2n)sum(-1)^(r) .^(2n)C_(r).^(2n)C_(2n-r)`
= Coefficient of `x^(2n)` in `(1-x)^(2n)(1+x)^(2n)`
= Coefficient of `x^(2n)` is `(1-x)^(2n)`
`= (-1)^(n).^(2n)C_(n)`
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