$$U_{665}(x, y) = \frac{1 - \sqrt{4 x \left(y + 1\right) + 1}}{2 x}$$

$$T_{665}(n, m, k) = \frac{\left(-1\right)^{k + n} k {\binom{k + n}{m}} {\binom{k + 2 n - 1}{k + n - 1}}}{k + n}$$