$$U_{569}(x, y) = \frac{\left(1 - y\right)^{6} \left(1 - \sqrt{- \frac{4 x}{\left(1 - y\right)^{8}} + 1}\right)}{2 x}$$

$$T_{569}(n, m, k) = \frac{k {\binom{k + 2 n - 1}{n}} {\binom{2 k + m + 8 n - 1}{m}}}{k + n}$$