$$U_{1023}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right) \left(x + 1\right)^{3}}{2 y}$$

$$T_{1023}(n, m, k) = \frac{k {\binom{3 k}{n}} {\binom{k + 2 m - 1}{m}}}{k + m}$$