$$U_{1212}(x, y) = \frac{32768 y^{36} \left(- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{4 y^{6}} - \sqrt{- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{2 y^{6}} + 1} + 1\right)^{3}}{x^{6} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{18}}$$

$$T_{1212}(n, m, k) = \frac{9 k n {\binom{6 k + 2 n}{n}} {\binom{2 m + 6 n}{m}}}{\left(3 k + n\right) \left(m + 3 n\right)}$$