$$U_{800}(x, y) = x y + x + \frac{\sqrt{4 x y + 4 x + 1}}{2} + \frac{1}{2}$$

$$T_{800}(n, m, k) = \begin{cases}\frac{2 \left(-1\right)^{n - 1} k {\binom{n}{m}} {\binom{- 2 k + 2 n - 1}{n - 1}}}{n}&\text{if n>0} ,\ \\0&\text{if n=0} \end{cases} $$