Not OP, but the reason is because that the values that you’re looking for tell you how to do the weighings, where the rows are weighings and the columns are the indices (it’s easier to see if you don’t translate from {0,1,2} to {-1,0,1}). Because of the problem you need the same number of balls on each side of the scale.

By choosing 7-19, omit 13, you finding the center values of the range 0-26. This is important because when you write the values in ternary each column has the same number of 0 and 2 digits.

This would not hold if you simply used the values 0-12, for example. Then the first column has eight 0’s and four 1’s, so it does not correspond to a valid weighting.

It’s important to note that this method doesn’t work for the problem described - it only works if you know that the odd ball out is heavier. You can see my other comment for a description of why, and a working solution.