$$U_{844}(x, y) = \frac{1 - \sqrt{- \frac{4 x}{1 - y} + 1}}{2 x \left(1 - y\right)}$$

$$T_{844}(n, m, k) = \frac{k {\binom{k + 2 n - 1}{n}} {\binom{2 k + m + n - 1}{m}}}{k + n}$$