$$U_{1004}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right)^{3} \left(x + 1\right)}{8 y^{3}}$$

$$T_{1004}(n, m, k) = \frac{3 k {\binom{k}{n}} {\binom{3 k + 2 m - 1}{m}}}{3 k + m}$$