$$U_{542}(x, y) = \frac{y^{2} \left(\sqrt{4 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{4}}{16 y^{8}}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}}\right)}{- 2 y - \sqrt{1 - 4 y} + 1}$$

$$T_{542}(n, m, k) = \begin{cases}1&\text{if n=0, m=0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0, n>0} ,\ \\\frac{k {\binom{2 k + 2 m}{m}}}{k + m}&\text{if n=0, m>0} ,\ \\\frac{2 k \left(k - 4 n\right) {\binom{k - n - 1}{n - 1}} {\binom{2 k + 2 m - 8 n - 1}{m - 1}}}{m n}&\text{if n>0, m>0} \end{cases} $$