$$U_{627}(x, y) = 2 x - \sqrt{4 x^{2} + \left(y + 1\right)^{4}}$$

$$T_{627}(n, m, k) = \begin{cases}\left(-1\right)^{k} {\binom{2 k}{m}}&\text{if n = 0} ,\ \\\frac{\left(-1\right)^{k + 1} \cdot 4^{n} k {\binom{2 k - 2 n}{m}} {\binom{- \frac{k}{2} + \frac{n}{2} - 1}{n - 1}}}{2 n}&\text{if n > 0} \end{cases} $$