Author 
Comment/Response 
Bill Simpson

09/21/12 1:16pm
From reading your explanation I am assuming there may be a couple of misunderstandings.
1: Sum[i1,{i,1,n}] == n*i1. Perhaps you meant Sum[i1,{i1,1,n}].
2: Mathematica evaluates all the arguments inside a function call and then evaluates that function. What this means is that it will separately each Sum independently of all the other sums. For example
In[1]:= Sum[i1,{i1,1,3}]*Sum[i2,{i2,i1,3}]*Sum[i3,{i3,i2,3}]
Out[1]= (3*(4+i1)*(3+i1)*(4+i2)*(3+i2))/2
and I do not believe that is what you want. I am guessing that you somehow want to evaluate each sum "inside" the other sum in some way so that each value of i1 in your first sum is used while calculating the second sum and the value of it in that sum is used while calculating the third sum.
At this point I am uncertain enough about what the result will be that I'll not include the code that I initially wrote.
If you would post again and show the manual process that you would use if you were going to calculate the desired product, with all the intermediate details so that readers can see where the numbers are coming from, and do this for n=1, n=2 and n=3 then I think someone would be able to write a few lines of code that would accomplish what you want to do.
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