$$U_{223}(x, y) = \frac{2 x^{2} y - 2 x - \sqrt{1 - 4 x} + 1}{2 x^{2}}$$

$$T_{223}(n, m, k) = \begin{cases}{\binom{k}{m}}&\text{if n=0} ,\ \\\frac{2 k {\binom{k - 1}{m}} {\binom{2 k - 2 m + 2 n - 1}{n - 1}}}{n} \end{cases} $$