$$U_{699}(x, y) = \frac{y^{2} \left(1 - \sqrt{- \frac{2 x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{y^{2}} + 1}\right)}{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}$$

$$T_{699}(n, m, k) = \begin{cases}1&\text{if n+m=0} ,\ \\\frac{k n {\binom{2 m + 2 n}{m}} {\binom{k + 2 n - 1}{n}}}{\left(k + n\right) \left(m + n\right)} \end{cases} $$