$$U_{429}(x, y) = \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{2 y^{6}}\right)}}{18} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{216 y^{6}}} + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{6 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{36 y^{4} \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{2 y^{6}}\right)}}{18} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{216 y^{6}}}}$$

$$T_{429}(n, m, k) = \begin{cases}0&\text{if n=0 and m=0} ,\ \\\frac{k {\binom{k - 2 n - 1}{n - 1}}}{n}&\text{if n>0 and m=0} ,\ \\\frac{2 k {\binom{k - 2 n - 1}{n}} {\binom{2 k + 2 m - 6 n - 1}{m - 1}}}{m} \end{cases} $$