$$U_{637}(x, y) = 2 x - \sqrt{4 x^{2} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}}}$$

$$T_{637}(n, m, k) = \begin{cases}\left(-1\right)^{k}&\text{if m = 0, n = 0} ,\ \\\frac{2 \left(-1\right)^{- k + n} 4^{n} k {\binom{\frac{k}{2} + \frac{n}{2} - 1}{n}} {\binom{2 k + 2 m - 2 n - 1}{m - 1}}}{m}&\text{if m > 0} ,\ \\\frac{\left(-1\right)^{k - 1} \cdot 4^{n} k {\binom{- \frac{k}{2} + \frac{n}{2} - 1}{n - 1}}}{2 n} \end{cases} $$