$$U_{413}(x, y) = x + \frac{\sqrt{4 y + 1}}{2} + \frac{1}{2}$$

$$T_{413}(n, m, k) = \begin{cases}1&\text{if n=0, m=0} ,\ \\{\binom{k}{n}}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{k - 1}{n}} {\binom{- k + 2 m + n - 1}{m - 1}}}{m} \end{cases} $$