$$U_{323}(x, y) = \frac{2 y + \sqrt{1 - 4 y} + \sqrt{2} \sqrt{4 x y^{3} + 2 y^{2} - 4 y + \sqrt{1 - 4 y} \left(- 4 x y^{3} + 2 y - 1\right) + 1} - 1}{2 y \sqrt{1 - 4 y} - 2 y}$$

$$T_{323}(n, m, k) = \begin{cases}0&\text{if n<k} ,\ \\\frac{\left(-1\right)^{n} k {\binom{- k + 2 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{n} \left(-1\right)^{m - 1} k \left(k - 3 n\right) {\binom{- k + 2 n - 1}{n - 1}} {\binom{- k - m + 3 n - 1}{m - 1}}}{m n} \end{cases} $$