$$U_{509}(x, y) = \frac{2 y + \sqrt{1 - 4 y} - 1}{y^{2} \left(\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{2 y^{4}} - 2\right)}$$

$$T_{509}(n, m, k) = \frac{k \left(k + 2 n\right) {\binom{k + n}{n}} {\binom{2 k + 2 m + 4 n}{m}}}{\left(k + n\right) \left(k + m + 2 n\right)}$$