$$U_{253}(x, y) = \frac{1}{\sqrt{- 4 x^{2} - 4 y^{2} + 1}}$$

$$T_{253}(n, m, k) = \begin{cases}\frac{\left(\left(-1\right)^{m} + 1\right) \left(\left(-1\right)^{n} + 1\right) {\binom{\frac{m}{2} + \frac{n}{2}}{\frac{n}{2}}} {\binom{\frac{k}{2} + \frac{m}{2} + \frac{n}{2} - \frac{1}{2}}{\frac{m}{2} + \frac{n}{2}}} {\binom{k + m + n - 1}{\frac{k}{2} + \frac{m}{2} + \frac{n}{2} - \frac{1}{2}}}}{4 {\binom{k - 1}{\frac{k}{2} - \frac{1}{2}}}}&\text{if k odd} ,\ \\\frac{\left(\left(-1\right)^{m} + 1\right) \left(\left(-1\right)^{n} + 1\right) {\binom{\frac{m}{2} + \frac{n}{2}}{\frac{n}{2}}} {\binom{\frac{k}{2} + \frac{m}{2} + \frac{n}{2}}{\frac{m}{2} + \frac{n}{2}}} {\binom{k + m + n}{\frac{k}{2} + \frac{m}{2} + \frac{n}{2}}}}{4 {\binom{k}{\frac{k}{2}}}}&\text{if k even} \end{cases} $$