$$U_{623}(x, y) = \frac{x^{2} + 2 x \left(1 - \frac{4 \sin^{2}{\left(\frac{\arcsin{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right) + \left(1 - \frac{4 \sin^{2}{\left(\frac{\arcsin{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}{1 - \frac{4 \sin^{2}{\left(\frac{\arcsin{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}}$$

$$T_{623}(n, m, k) = \begin{cases}{\binom{2 k}{n}}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} \left(k - n\right) {\binom{2 k}{n}} {\binom{- k + 3 m + n - 1}{m - 1}}}{m}&\text{if m>0} \end{cases} $$