Your first iterator has the form {1, n-6}. The proper form of an iterator is something like {a, 1, n}, that is,

(variable, start_value, end_value}

The variable needs to be symbolic. The start and end values may be numeric or symbolic expressions. There is a short form for iterators when the start value is 1: {a, n} is equivalent to {a, 1, n}. All of your iterators lack a variable.

When you fix that, you will probably discover another syntax issue: abcdef is a single variable, whereas a b c d e f is a product of six variables. It is important to put spaces between symbol names. My guess is that you want the product.

Sir, now it works. But it's monstruous. I try to find a numerical rule of these sums: 1x2x...x(k-1)xk+1x2x...x(k-1)x(k+1)+...+1x2x...x(k-1)xn+1x2x...xkx(k+1)+...+(n-k+1)x(n-k+2)x...xn where n is natural, greater than or equal with k and k is natural, greater than 1 (each sum is fixed for a k). Until now I've tried with simple form of Wolfram Alpha, for users (cases k=1,2,3,4,5,6 which means at most five nested sums), but now with the increased computational power of Wolfram Mathematica I hope to calculate these kind of summations for bigger values of k. The idea with these sums is as it follows: there appears some arithmetical progressions that can be calculated and multiplied with each sum. E.g.: k=2 1x2+1x3+...+1xn+2x3+2x4+...+2xn+...+(n-1)xn k=3 1x2x3+1x2x4+...+1x2xn+1x3x4+1x3x5+...+1x3xn+...+(n-2)x(n-1)xn.

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
*)

And what shall I write correctly? Can you send me here the text modified please?