$$U_{1005}(x, y) = \frac{\left(x + 1\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}}$$

$$T_{1005}(n, m, k) = \frac{2 k {\binom{k}{n}} {\binom{4 k + 2 m}{m}}}{2 k + m}$$