$$U_{157}(x, y) = \frac{\sqrt{4 x y^{2} + 8 x y + 4 x + 1}}{2} + \frac{1}{2}$$

$$T_{157}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\0&\text{if n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{2 n}{m}} {\binom{- k + 2 n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$