$$U_{439}(x, y) = x + \frac{y}{2} + \frac{\sqrt{4 x \left(y + 1\right) + \left(y + 1\right)^{2}}}{2} + \frac{1}{2}$$

$$T_{439}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{2 \left(-1\right)^{n - 1} k {\binom{- 2 k + 2 n - 1}{n - 1}}}{n}&\text{if n>0 and m=0} ,\ \\\frac{k {\binom{k - n - 1}{m - 1}} {\binom{2 k - n - 1}{n}}}{m} \end{cases} $$