$$U_{1112}(x, y) = \left(\frac{x \left(1 - \sqrt{1 - 4 y}\right)}{2 y} + 1\right)^{2}$$

$$T_{1112}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{n {\binom{2 k}{n}} {\binom{2 m + n - 1}{m}}}{m + n} \end{cases} $$