$$U_{407}(x, y) = \sqrt[3]{\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{4 y^{2}} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}}\right)} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{36 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{216 y^{6}}} + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{6 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{36 y^{4} \sqrt[3]{\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{4 y^{2}} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}}\right)} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{36 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{216 y^{6}}}}$$

$$T_{407}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\left(-1\right)^{n - 1} k {\binom{- k + 3 n - 1}{n - 1}}&\text{if m=0} ,\ \\\frac{2 k {\binom{k - 2 n}{n}} {\binom{2 k + 2 m - 4 n - 1}{m - 1}}}{m}&\text{if m>0} \end{cases} $$