$$U_{478}(x, y) = \frac{\sqrt{3} \sqrt{y} \left(\sqrt{\frac{8 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + \frac{16 \sin^{4}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{2}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)}{4 \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$

$$T_{478}(n, m, k) = \begin{cases}\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if n>0 and m=0} ,\ \\\frac{k {\binom{k + 3 m - 1}{m}}}{k + 2 m}&\text{if n=0 and m>0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - 3 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k - 2 m + 3 n - 1}{m - 1}}}{m n}&\text{if n>0 and m>0} ,\ \\1&\text{if n=0 and m=0} \end{cases} $$