$$U_{1167}(x, y) = \frac{16 y^{12} \left(1 - \sqrt{- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{2 y^{6}} + 1}\right)^{2}}{x^{2} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{6}}$$

$$T_{1167}(n, m, k) = \frac{6 k n {\binom{2 m + 6 n}{m}} {\binom{2 k + 2 n - 1}{n}}}{\left(2 k + n\right) \left(m + 3 n\right)}$$