$$U_{1221}(x, y) = \frac{\sqrt{2} \sqrt{\frac{1 - \sqrt{1 - 16 y}}{y}}}{4 \left(1 - x\right)^{3}}$$

$$T_{1221}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{k {\binom{\frac{k}{2} + 2 m - \frac{1}{2}}{m}} {\binom{k + 4 m - 1}{\frac{k}{2} + 2 m - \frac{1}{2}}} {\binom{3 k + n - 1}{n}}}{\left(k + 2 m\right) {\binom{k + 2 m - 1}{\frac{k}{2} + m - \frac{1}{2}}}}&\text{if k odd} ,\ \\\frac{4^{m} k {\binom{\frac{k}{2} + 2 m - 1}{m}} {\binom{3 k + n - 1}{n}}}{k + 2 m}&\text{if k even} \end{cases} $$