$$U_{1224}(x, y) = \frac{\frac{\sin{\left(\frac{\operatorname{asin}{\left(216 y^{2} - 1 \right)}}{3} \right)}}{6 y} + \frac{1}{12 y}}{\left(1 - x\right)^{3}}$$

$$T_{1224}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{\frac{k}{2} + \frac{3 m}{2} - 1}{m}} {\binom{3 k + n - 1}{n}}}{k + m}&\text{if k even} ,\ \\\frac{k {\binom{\frac{k}{2} + \frac{3 m}{2} - \frac{1}{2}}{m}} {\binom{k + 3 m - 1}{\frac{k}{2} + \frac{3 m}{2} - \frac{1}{2}}} {\binom{3 k + n - 1}{n}}}{\left(k + m\right) {\binom{k + m - 1}{\frac{k}{2} + \frac{m}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$