$$U_{669}(x, y) = \frac{\sqrt{\frac{4 x}{1 - y} + 1}}{2} + \frac{1}{2}$$

$$T_{669}(n, m, k) = \begin{cases}{\binom{m + 2 n - 1}{m}}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 2 n - 1}{n - 1}} {\binom{m + n - 1}{m}}}{n}&\text{if n>0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 2 n - 1}{n - 1}} {\binom{m + 2 n - 1}{m}}}{n} \end{cases} $$