$$U_{1000}(x, y) = \frac{2 y}{\left(1 - \sqrt{1 - 4 y}\right) \left(- \frac{x \left(1 - \sqrt{1 - 4 y}\right)}{2 y} + 1\right)}$$

$$T_{1000}(n, m, k) = \begin{cases}{\binom{k + n - 1}{n}}&\text{if m=0 } ,\ \\\frac{\left(- k + n\right) {\binom{k + n - 1}{n}} {\binom{- k + 2 m + n - 1}{m - 1}}}{m}&\text{if m>0} \end{cases} $$