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 (^(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_n=(2^n[1. 3. 5 (2n-1)])/(n !)

Prove that : C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0

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

Prove that: .^(2)C_(2)+^(3)C_(2)+^(4)C_(2)+…..+^(n+1)C_(2)=1/6n(n+1)(n+2)

Prove that .^n C_0 . ^(2n) C_n- ^n C_1 . ^(2n-2)C_n+^n C_2 . ^(2n-4)C_n-=2^ndot

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

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: .^(2)C_(2)+^(3)C_(2)+^(4)C_(2)+…..+^(n+1)C_(2)=1/6n(n+1)(n+2)

Prove that "^n C_0^(2n)C_n-^n C_1^(2n-1)C_n+^n C_2xx^(2n-2)C_n++(-1)^n^n C_n^n C_n=1.