$$U_{1215}(x, y) = \frac{\left(\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}} + 1\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}}$$

$$T_{1215}(n, m, k) = \frac{\left(2 k + 2 n\right) {\binom{k}{n}} {\binom{4 k + 2 m + 4 n}{m}}}{2 k + m + 2 n}$$