$$U_{516}(x, y) = \frac{2 \sqrt{3} \left(y + 1\right)^{2} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{x}}{2} \right)}}{3} \right)}}{3 \sqrt{x}}$$

$$T_{516}(n, m, k) = \frac{k {\binom{2 k}{m}} {\binom{k + 3 n - 1}{n}}}{k + 2 n}$$