$$U_{63}(x, y) = \frac{1 - \sqrt{1 - 4 x}}{2 x + y \left(\sqrt{1 - 4 x} - 1\right)}$$

$$T_{63}(n, m, k) = \frac{k {\binom{k + m}{m}} {\binom{k + m + 2 n - 1}{n}}}{k + m + n}$$