$$U_{823}(x, y) = \left(\frac{2 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + 1\right)^{2}$$

$$T_{823}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{n {\binom{2 k}{n}} {\binom{3 m + n - 1}{m}}}{2 m + n} \end{cases} $$