$$U_{410}(x, y) = \frac{\sqrt{4 y + 1}}{4} + \frac{\sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{1}{4}$$

$$T_{410}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0,n>0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{k - n}{n}} {\binom{- k + 2 m + n - 1}{m - 1}}}{m} \end{cases} $$