$$U_{1117}(x, y) = \left(\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6}} + 1\right)^{2}$$

$$T_{1117}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{3 n {\binom{2 k}{n}} {\binom{2 m + 6 n}{m}}}{m + 3 n} \end{cases} $$