$$U_{814}(x, y) = \frac{2 \sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{x} \left(y + 1\right)}{2} \right)}}{3} \right)}}{3 \sqrt{x} \left(y + 1\right)}$$

$$T_{814}(n, m, k) = \frac{k {\binom{2 n}{m}} {\binom{k + 3 n - 1}{n}}}{k + 2 n}$$