Find the sum `sum_(j=0)^n( ^(4n+1)C_j+^(4n+1)C_(2n-j))` .

Find the sum sum_(r=0) .^(n+r)C_r .

Find the sum sum_(i=0)^r.^(n_1)C_(r-i) .^(n_2)C_i .

Find the sum sum_(r=1)^(n) r^(2) (""^(n)C_(r))/(""^(n)C_(r-1)) .

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

Find the sum (sumsum)_(0leiltjlen) ""^(n)C_(i).""^(n)C_(j) .

Find the sum_(r =0)^(r) ""^(n_(1))C_((r-i))""^(n_(2))C_(i) .

Find the following sum: sumsum_(i ne j) ""^(n)C_(i).""^(n)C_(j)

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)^2dot

Find the following sums : (i) .^(n)C_(0)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(6)+"....." (ii) .^(n)C_(1)-.^(n)C_(3)+.^(n)C_(5)-.^(n)C_(7)+"...." (iii) .^(n)C_(0)+.^(n)C_(4)+.^(n)C_(8)+.^(n)C_(12)+"....." (iv) .^(n)C_(2) + .^(n)C_(6) + .^(n)C_(10)+.^(n)C_(14)+"......" (v) .^(n)C_(1) + .^(n)C_(5)+.^(n)C_(9)+.^(n)C_(13)+"...." (vi) .^(n)C_(3) + .^(n)C_(7) + .^(n)C_(11) + .^(n)C_(15) + "....."

If n in N such that is not a multiple of 3 and (1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r ) , then find the value of sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r ) .