$$U_{1039}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6} \left(1 - x\right)^{2}}$$

$$T_{1039}(n, m, k) = \frac{3 k {\binom{6 k + 2 m}{m}} {\binom{2 k + n - 1}{n}}}{3 k + m}$$