$$U_{577}(x, y) = \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6}} + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{2 y^{2}}$$

$$T_{577}(n, m, k) = \frac{\left(k + 2 n\right) {\binom{k}{n}} {\binom{2 k + 2 m + 4 n}{m}}}{k + m + 2 n}$$