$$U_{1031}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right)^{3}}{y^{3} \left(8 - 8 x\right)}$$

$$T_{1031}(n, m, k) = \frac{3 k {\binom{k + n - 1}{n}} {\binom{3 k + 2 m - 1}{m}}}{3 k + m}$$