$$U_{507}(x, y) = \frac{\left(\sqrt{4 x + 1} + 1\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{4 y^{2}}$$

$$T_{507}(n, m, k) = \begin{cases}\frac{k {\binom{2 k + 2 m}{m}}}{k + m}&\text{if n==0} ,\ \\\frac{\left(-1\right)^{n - 1} k^{2} {\binom{2 k + 2 m}{m}} {\binom{- k + 2 n - 1}{n - 1}}}{n \left(k + m\right)}&\text{if n>0} \end{cases} $$