$$U_{510}(x, y) = \frac{y^{4} \left(2 - 2 \sqrt{- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{2 y^{6}} + 1}\right)}{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}$$

$$T_{510}(n, m, k) = \frac{k \left(k + 3 n\right) {\binom{k + 2 n}{n}} {\binom{2 k + 2 m + 6 n}{m}}}{\left(k + 2 n\right) \left(k + m + 3 n\right)}$$