$$U_{1178}(x, y) = \frac{8 y^{12} \left(1 - \sqrt{- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}} + 1}\right)^{3}}{x^{3} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{6}}$$

$$T_{1178}(n, m, k) = \frac{6 k n {\binom{2 m + 4 n}{m}} {\binom{3 k + 2 n - 1}{n}}}{\left(3 k + n\right) \left(m + 2 n\right)}$$