I don't understand what you want to compute. Here's simple modification of your original code to include the variables. The result factors somewhat nicely and does not seem monstrous to me, but maybe it does to you:
Factor[
Sum[1/2 a b c d e f (e + f) (f - e - 1), {a, 1, n - 6}, {b, a + 1,
n - 5}, {c, b + 1, n - 4}, {d, c + 1, n - 3}, {e, d + 1,
n - 2}, {f, e + 2, n}]
]
(*
((-6 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) n^2 (1 +
n)^2 (80 + 34 n - 57 n^2 - 18 n^3 + 9 n^4))/5806080
*)