$$U_{1253}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6} \left(- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6}} + 1\right)^{2}}$$

$$T_{1253}(n, m, k) = \frac{\left(3 k + 3 n\right) {\binom{2 k + n - 1}{n}} {\binom{6 k + 2 m + 6 n}{m}}}{3 k + m + 3 n}$$