$$U_{456}(x, y) = \frac{y}{3} + \frac{\left(y + 1\right)^{2}}{9 \sqrt[3]{\frac{x}{2 y + 2} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{4}\right)}}{18 y + 18} + \frac{\left(y + 1\right)^{3}}{27}}} + \sqrt[3]{\frac{x}{2 y + 2} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{4}\right)}}{18 y + 18} + \frac{\left(y + 1\right)^{3}}{27}} + \frac{1}{3}$$

$$T_{456}(n, m, k) = \begin{cases}0&\text{if n=0 and m=0} ,\ \\{\binom{k}{m}}&\text{if m>0} ,\ \\\frac{k {\binom{k - 4 n}{m}} {\binom{k - 2 n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$