$$U_{683}(x, y) = \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{2 y^{2}} + 1$$

$$T_{683}(n, m, k) = \begin{cases}1&\text{if n+m=0} ,\ \\\frac{n {\binom{k}{n}} {\binom{2 m + 2 n}{m}}}{m + n} \end{cases} $$