$$U_{1038}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4} \left(1 - x\right)^{2}}$$

$$T_{1038}(n, m, k) = \frac{2 k {\binom{4 k + 2 m}{m}} {\binom{2 k + n - 1}{n}}}{2 k + m}$$