Prove that `(^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot`

Prove that (""^(2n)C_(0))^2-(""^(2n)C_(1))^2+(""^(2n)C_(2))^2-.....+(-1)^n(""^(2n)C_(2n))^2=(-1)^n.""^(2n)C_(n)

Prove that ^nC_(0)^(n)C_(0)-^(n+1)C_(1)^(n)C_(1)+^(n+2)C_(2)^(n)C_(2)-...=(-1)^(n)

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-2)C_(n)+^(n)C_(2)^(2n-4)C_(n)-...=2^(n)

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-1)C_(n)+^(n)C_(2)xx^(2n-2)C_(n)++(-1)^(n)sim nC_(n)^(n)C_(n)=1

Prove that: ^(2n)C_0-3.^(2n)C_1+3^2.^(2n)C_2-..+(-1)^(2n) ..3^(2n)^(2n)C_(2n)=4^n for all value of N

Prove that sum_(r=0)^(2n)(.^(2n)C_(r))^(2)=n^(4n)C_(2n)

If (.^(2n)C_1)^2+ 2.(.^(2n)C_2)^2+3.(.^(2n)C_3)^2+...+2n. (.^(2n)C_(2n))^2 = 18 .^(4n-1)C_(2n-1)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) + C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" + (.^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .