$$U_{691}(x, y) = \frac{1 - \sqrt{\frac{8 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + 1}}{2 x}$$

$$T_{691}(n, m, k) = \begin{cases}\left(-1\right)^{k}&\text{if m+n=0} ,\ \\\frac{\left(-1\right)^{- k + n} k {\binom{k + 2 n - 1}{k + n - 1}} {\binom{k + 3 m + n - 1}{m}}}{k + 2 m + n}&\text{if n+m>0} \end{cases} $$