$$U_{578}(x, y) = \frac{2 y + \sqrt{1 - 4 y} - 1}{y^{2} \left(\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{4 y^{6}} - 2\right)}$$

$$T_{578}(n, m, k) = \frac{\left(k + 3 n\right) {\binom{k + n - 1}{n}} {\binom{2 k + 2 m + 6 n}{m}}}{k + m + 3 n}$$