$$U_{1223}(x, y) = \sqrt{4 x + 1} \left(y + 1\right)^{3}$$

$$T_{1223}(n, m, k) = \begin{cases}1&\text{if n=0, m=0, k=0} ,\ \\\frac{4^{m} k {\binom{\frac{k}{2} + \frac{3 m}{2} - 1}{m}} {\binom{2 k + n - 1}{n}}}{k + m}&\text{if k even} ,\ \\\frac{k {\binom{\frac{k}{2} + \frac{3 m}{2} - \frac{1}{2}}{m}} {\binom{k + 3 m - 1}{\frac{k}{2} + \frac{3 m}{2} - \frac{1}{2}}} {\binom{2 k + n - 1}{n}}}{\left(k + m\right) {\binom{k + m - 1}{\frac{k}{2} + \frac{m}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$