$$U_{477}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}{9 \sqrt[3]{\frac{x}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}\right)}}{18 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)} + \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{3}}{27}}} + \sqrt[3]{\frac{x}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}\right)}}{18 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)} + \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{3}}{27}} - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{9} + \frac{1}{3}$$

$$T_{477}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{k {\binom{k - 2 n - 1}{n - 1}}}{n}&\text{if m=0 and n>0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + 3 m + n - 1}{m - 1}}}{m}&\text{if n=0 and m>0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - 4 n\right) {\binom{k - 2 n - 1}{n - 1}} {\binom{- k + 3 m + 4 n - 1}{m - 1}}}{m n}&\text{if m>0 and n>0} \end{cases} $$