Show that `C_1^2-2C_2^2+3.C_3^2- ……….-2n.C_(2n)^2=(-1)^(n-1).n.C_n` where `C_r` stands for ` ''^(2n)C_r''`

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

Prove that C_0-2^2C_1+3^2C_2-4^2C_3++(-1)^n(n+1)^2xxC_n=0w h e r eC_r=^n C_r .

Let n be a positive integer and (1+x)^(n)+C_(0)+C_(1)x+C_(2)x^(2)+C_(3)x^(3)+ . . .+C_(r)x^(r)+ . . .+C_(n-1)x^(n-1)+C_(n)x^(n) Where C_(r) stands for .^(n)C_(r) , then Q. The values of underset(r=0)overset(n)(sum)underset(s=0)overset(n)(sum)(C_(r)+C_(S)) is

Let n be a positive integer and (1+x)^(n)+C_(0)+C_(1)x+C_(2)x^(2)+C_(3)x^(3)+ . . .+C_(r)x^(r)+ . . .+C_(n-1)x^(n-1)+C_(n)x^(n) Where C_(r) stands for .^(n)C_(r) , then Q. The value of underset(r=0)overset(n)(sum)underset(s=0)overset(n)(sum),C_(r),C_(S) is

Prove that C_3 + 2.C_4+ 3.C_5 + ……..+ (n-2).C_n = (n-4).2^(n-1) + (n+2) where n > 3

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 (n-1)^(2) C_(1) + (n-3)^(2) C_(3) + (n-5)^(2) C_(5) . + ….= n (n+1)2^(n-3) , where C_(r) stands for ""^(n)C_(r) .

.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)=

Show that (C_(0))/(1) - (C_(1))/(4) + (C_(2))/(7) - … + (-1)^(n) (C_(n))/(3n +1) = (3^(n) * n!)/(1*4*7…(3n+1)) , where C_(r) stands for ""^(n)C_(r) .

If (1+x)^n=underset(r=0)overset(n)C_(r)x^r then prove that C_(1)^2+2.C_(2)^(2)+3.C_(3)^2 +…….+n.C_(n)^(2)=((2n-1)!/((n-1)!)^2