If for `n in N, sum_(k=0)^(2n) (-1)^(k)(""^(2n)C_(k))^(2) = A`, then find the value of `sum_(k=0)^(2n) (-1)^(k) (k = 2n)(""^(2n)C_(k))^(2)`.

Explanation

Let
`S=underset(k=0)overset(2n)sum(-1)^(k)(k-2n)(.^(2n)C_(k))^(2)=-underset(k=0)overset(2n)sum(-1)^(k)(2n-k)(.^(2n)C_(k))^(2)`
`:. S =-(2n(.^(2n)C_(0))^(2)-(2n-1)(.^(2n)C_(1))^(2)+(2n-2)(.^(2n)C_(2))^(2)-"......."-1(.^(2n)C_(2n-1))^(2)+0(.^(2n)C_(2n))^(2))" "(1)`
or `S = -(2n(.^(2n)C_(2n))^(2)-(2n-1)(.^(2n)C_(2n-1))^(2)+(2n-2)(.^(2n)C_(2n-2))^(2)-"........"-1(.^(2n)C_(1))^(2)+0(.^(2n)C_(0))^(2))`
or `S = - (0(.^(2n)C_(0))^(2)-1(.^(2n)C_(1))^(2) + 2(.^(2n)C_(2))^(2)-"......"-(2n-1) xx (.^(2n)C_(2n-1))^(2)+2(.^(2n)C_(2n))^(2))" "(2)`
Adding (1) and (2), we get
`2S = -2n[(.^(2n)C_(0))^(2)-(.^(2n)C_(1))^(2)+(.^(2n)C_(2))^(2)-(.^(2n)C_(3))^(2)+"....."-(.^(2n)C_(2n-1))^(2)+(.^(2n)C_(2n))^(2)]`
or `2S = -nA`