Another general observation is that the product of each row is one. This follows from the observations that the rows are bookended by \\(\frac{1}{n}\\) and \\(\frac{n}{1}\\), and the fact that \\(f(\frac{p}{q}) = \frac{q}{p + q - 2(p \mod q)}\\). I suppose that to complete the proof, you'd need to show that the RHS is still in lowest terms whenever the LHS is. But if \\((p, q) = 1\\), then so is \\( (q, p + q - 2r) \\), where \\(r = p \mod q\\), so we're ok.