$$U_{1291}(x, y) = \frac{\sqrt{2} \sqrt{\frac{1 - \sqrt{1 - 16 y}}{y}}}{4 - 4 x}$$

$$T_{1291}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0, k=0} ,\ \\\frac{4^{m} k {\binom{\frac{k}{2} + 2 m - 1}{m}}}{k + 2 m}&\text{if k even} ,\ \\\frac{k {\binom{\frac{k}{2} + 2 m - \frac{1}{2}}{m}} {\binom{k + 4 m - 1}{\frac{k}{2} + 2 m - \frac{1}{2}}}}{\left(k + 2 m\right) {\binom{k + 2 m - 1}{\frac{k}{2} + m - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$