$$U_{506}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right) \left(\sqrt{4 x + 1} + 1\right)}{4 y}$$

$$T_{506}(n, m, k) = \begin{cases}\frac{k {\binom{k + 2 m - 1}{m}}}{k + m}&\text{if n==0} ,\ \\\frac{\left(-1\right)^{n - 1} k^{2} {\binom{- k + 2 n - 1}{n - 1}} {\binom{k + 2 m - 1}{m}}}{n \left(k + m\right)}&\text{if n>0} \end{cases} $$