$$U_{347}(x, y) = \frac{2 y^{2} \left(x + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{2 y^{2}}\right)^{2}}{- 2 y - \sqrt{1 - 4 y} + 1}$$

$$T_{347}(n, m, k) = \begin{cases}{\binom{2 k}{n}}&\text{if m=0} ,\ \\\frac{2 \left(k - n\right) {\binom{2 k}{n}} {\binom{2 k + 2 m - 2 n - 1}{m - 1}}}{m} \end{cases} $$