$$U_{559}(x, y) = \frac{1 - \sqrt{- 4 x \left(y + 1\right)^{8} + 1}}{2 x \left(y + 1\right)^{6}}$$

$$T_{559}(n, m, k) = \frac{k {\binom{k + 2 n}{n}} {\binom{2 k + 8 n}{m}}}{k + 2 n}$$