You don't need to find the values of a_n here. The question asks about the sum from r=2 to 2n of a(r)(-1)^(r)r(r-1)/2

If f(x) = the sum from r=0 to 2n of a(r)x^(r) then

f'(x) = the sum from r=1 to 2n of ra(r)x^(r-1) and

f''(x) = the sum from r=2 to 2n of r(r-1)a(r)x^(r-2)

Therefore f''(-1)/2 = the sum from r=2 to 2n of r(r-1)a(r)(-1)^(r-2)/2 which is the same as what you want.

We're given that f(x) = (1+x+x^(2))^(n)

If you calculate f''(x) and plug in x=-1 you'll get n(n+1). Dividing by 2 gives you the answer.