$$U_{111}(x, y) = \frac{2}{x + \sqrt{x^{2} - 2 x - 4 y + 1} + 1}$$

$$T_{111}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{k {\binom{m + n}{m}} {\binom{k + 2 m + n - 1}{m - 1}}}{m + n} \end{cases} $$