$$U_{1226}(x, y) = \left(x + 1\right)^{2} \left(2 y + \sqrt{4 y^{2} + 1}\right)$$

$$T_{1226}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{2 k}{n}} {\binom{\frac{k}{2} + \frac{m}{2}}{m}}}{k + m},\ \\\frac{k {\binom{2 m}{m}} {\binom{k + m + 1}{2 m}}}{\left(k + m\right) {\binom{\frac{k}{2} + \frac{m}{2} + \frac{1}{2}}{m}}},\ \\\frac{\left(-1\right)^{- \frac{k}{2} + \frac{m}{2} - \frac{1}{2}} {\binom{m}{\frac{k}{2} + \frac{m}{2} + \frac{1}{2}}} {\binom{2 m}{m}}}{{\binom{2 m}{k + m + 1}}} \end{cases} $$