Prove that `(2nC_1)^2+2.(2nC_2)^2+3.(2nC_3)^2+.....+2n.(2nC_(2n))^2=((4n-1)!)/([(2n-1)!])^2`

Prove that (2nC_0)^2-(2nC_1)^2+(2nC_2)^2+.....+(2nC_2n)^2=(-1)^n2nC_n

Prove that (2nC_0)^2+(2nC_1)^2+(2nC_2)^2-+(2nC_(2n))^2=(-1)^n2nC_ndot

c_(1)^(2)+2C_(2)^(2)+3C_(3)^(2)+....+nC_(n)^(2)=((2n-1)!)/ ([(n-1)!^(2)))

(nC_(0))^(2)+(nC_(1))^(2)+(nC_(2))^(2)+......+(nC_(n))^(2)

Prove that nC_1(nC_2)^2(n C_3)^3.......(n C_n)^n le ((2^n)/(n+1))^((n+1)C_2),AAn in Ndot

Show that C_1^2 + 2C_2^2 + 3(3^2 + ... + "^nC_n^2 = ((2n-1!))/{(n-1)!}^2

(nC_(0))^(2)-(nC_(1))^(2)+(nC_(2))^(2)+....+(-1)^(n)(nC_(n))^(2)

Prove that ^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n

Given that C_(1)+2C_(2)x+3C_(3)x^(2)+...+2nC_(2n)x^(2n-1)=2n(1+x)^(2n-1),whereC_(r)=(2n)!/[r!(2n-r)!];r=0,1,2 then prove that C_(1)^(2)-2C_(2)^(2)+3C_(3)^(2)-...-2nC_(2n)^(2)=(-1)^(n)nC_(n).

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