$$U_{427}(x, y) = \frac{\left(\frac{x}{1 - y} + 1\right)^{3}}{\left(1 - y\right)^{2}}$$

$$T_{427}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if n>0 and m=0} ,\ \\\frac{2 k {\binom{k - n - 1}{n}} {\binom{2 k + 2 m - 4 n - 1}{m - 1}}}{m} \end{cases} $$