$$U_{813}(x, y) = \frac{2 \sqrt{3} \left(1 - y\right) \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{x}}{2 - 2 y} \right)}}{3} \right)}}{3 \sqrt{x}}$$

$$T_{813}(n, m, k) = \frac{k {\binom{k + 3 n - 1}{n}} {\binom{m + 2 n - 1}{m}}}{k + 2 n}$$