$$U_{383}(x, y) = \frac{\sqrt{\frac{4 x}{1 - y} + \frac{1}{\left(1 - y\right)^{2}}}}{2} + \frac{1}{2 - 2 y}$$

$$T_{383}(n, m, k) = \begin{cases}{\binom{k + m - 1}{m}}&\text{if n=0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}} {\binom{k + m - n - 1}{m}}}{n}&\text{if n>0} \end{cases} $$