$$U_{563}(x, y) = - \frac{1}{\left(1 - y\right) \left(\frac{x}{\left(1 - y\right)^{3}} - 1\right)}$$

$$T_{563}(n, m, k) = {\binom{k + n - 1}{n}} {\binom{k + m + 3 n - 1}{m}}$$