How to make i not euqal to j in a Double Sum

Question

I want to compute a double sum like below with i != j. (When i =j, the denominator will be zero, which can not be assessed). Does anyone know how to do it? I want to skip the calculation when i=j.

Sum[1/(i-j),{i,1,10},{j,1,10}]

I found a appraoch like this:

pairs = Subsets[Range[10], {2}];
test[{i_, j_}] := 1/(i - j);
Total[test /@ datas]*2

which gives a result of -(4861/126), which is the anwser I need. But this method requires more time than the Sum funciton in Mathematica. (In addition, my function is more complex than the 1/i-j.

I would like to bid my sincere appreciate to any help.

Answer 1

m = SparseArray[{{i_, j_} /; j > i -> 1/(i - j)}, {10, 10}];
Total@Total@Normal[m]*2

Answer 2

Just because of its symmetry your sum is equal to zero. If you need sometjing else, where you need to exclude the case i=j under the sum, you may do as follows:

Sum[If[i != j, 1/(1 + (i - j)^2), 0], {i, 1, 3}, {j, 1, 3}]

(* 12/5 *)

Have fun!