$$U_{1001}(x, y) = \frac{4 y^{2}}{\left(1 - \sqrt{1 - 4 y}\right)^{2} \left(- \frac{x \left(1 - \sqrt{1 - 4 y}\right)}{2 y} + 1\right)}$$

$$T_{1001}(n, m, k) = \begin{cases}{\binom{k + n - 1}{n}}&\text{if m=0} ,\ \\\frac{\left(- 2 k + n\right) {\binom{k + n - 1}{n}} {\binom{- 2 k + 2 m + n - 1}{m - 1}}}{m}&\text{if m>0} \end{cases} $$