$$U_{581}(x, y) = \frac{\sqrt{\frac{4 x}{\left(1 - 4 y\right)^{\frac{3}{2}}} + \frac{1}{1 - 4 y}}}{2} + \frac{1}{2 \sqrt{1 - 4 y}}$$

$$T_{581}(n, m, k) = \begin{cases}\frac{4^{m} k {\binom{\frac{k}{2} + m + \frac{n}{2} - 1}{m}}}{k + n}&\text{if n==0} ,\ \\\frac{\left(-1\right)^{n - 1} \cdot 4^{m} k {\binom{- k + 2 n - 1}{n - 1}} {\binom{\frac{k}{2} + m + \frac{n}{2} - 1}{m}}}{n}&\text{if n>0} \end{cases} $$