$$U_{812}(x, y) = \frac{2 \sqrt{3} \left(x + y\right) \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{x}}{2} \right)}}{3} \right)}}{3 \sqrt{x}}$$

$$T_{812}(n, m, k) = \begin{cases}\frac{k {\binom{k}{m}} {\binom{- 2 k + 3 m + 3 n - 1}{- k + m + n}}}{- k + 2 m + 2 n},\ \\0&\text{if m=0,n=0} \end{cases} $$