$$U_{1047}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right)^{2}}{4 y^{2} \left(1 - x\right)^{3}}$$

$$T_{1047}(n, m, k) = \frac{2 k {\binom{2 k + 2 m - 1}{m}} {\binom{3 k + n - 1}{n}}}{2 k + m}$$