$$U_{688}(x, y) = \frac{1 - \sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + 1}}{2 x}$$

$$T_{688}(n, m, k) = \begin{cases}\frac{\left(-1\right)^{- k + n} k {\binom{k + 2 n - 1}{k + n - 1}}}{k + n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{- k + n} \left(-1\right)^{m - 1} k {\binom{k + 2 n - 1}{k + n - 1}} {\binom{- k + 2 m - n - 1}{m - 1}}}{m} \end{cases} $$