$$U_{419}(x, y) = \frac{- y \left(2 - 2 x\right) - \sqrt{y \left(4 - 4 x\right) + 1} - 1}{2 x - 2}$$

$$T_{419}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{2 \left(-1\right)^{m - 1} k {\binom{- 2 k + 2 m - 1}{m - 1}}}{m}&\text{if m>0 and n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + m - 1}{n - 1}} {\binom{2 k - m - 1}{m}}}{n}&\text{if n>0} \end{cases} $$