$$U_{1006}(x, y) = \frac{\left(x + 1\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6}}$$

$$T_{1006}(n, m, k) = \frac{3 k {\binom{k}{n}} {\binom{6 k + 2 m}{m}}}{3 k + m}$$