$$U_{416}(x, y) = \frac{\sqrt{4 y + 1}}{6} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{36 \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{2}\right)}}{18} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}}} + \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{2}\right)}}{18} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}} + \frac{1}{6}$$

$$T_{416}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{k {\binom{k - 2 n - 1}{n - 1}}}{n}&\text{if n>0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{k - 2 n - 1}{n}} {\binom{- k + 2 m + 3 n - 1}{m - 1}}}{m} \end{cases} $$