$$U_{1216}(x, y) = \frac{\left(\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6}} + 1\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6}}$$

$$T_{1216}(n, m, k) = \frac{\left(3 k + 3 n\right) {\binom{k}{n}} {\binom{6 k + 2 m + 6 n}{m}}}{3 k + m + 3 n}$$