$$U_{824}(x, y) = \frac{- 4 \sqrt{3} x \sqrt{y} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)} + 3 y - \sqrt{- 24 \sqrt{3} x y^{\frac{3}{2}} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)} + 9 y^{2}}}{8 x^{2} \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$

$$T_{824}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{k n {\binom{2 k + 2 n}{n}} {\binom{3 m + n - 1}{m}}}{\left(k + n\right) \left(2 m + n\right)} \end{cases} $$