$$U_{47}(x, y) = \frac{y + \sqrt{- 4 x + y^{2} + y \left(4 x - 2\right) + 1} - 1}{2 x \left(y - 1\right)}$$

$$T_{47}(n, m, k) = \frac{k {\binom{k + m + n}{m}} {\binom{k + 2 n - 1}{n}}}{k + m + n}$$