The variable in symsum() cannot be used to index anything. You need to construct the vector of individual values and then sum() them.
For any individual k, how do you decide whether to add or subtract the 1/5* term? Or do you do both, thereby constructing 2^20 entries for Sj (all the possible combinations of plus and minus) ?
Your equations are for Sj but they do not involve j at any point: instead they use the undefined i .
I notice that both of your equations define Sj . Is the implication that the two Sj must equal each other?