$$U_{762}(x, y) = \frac{\left(1 - \sqrt{- 2 x \left(\sqrt{4 y + 1} + 1\right) + 1}\right)^{3}}{2 x^{3} \left(\sqrt{4 y + 1} + 1\right)^{2}}$$

$$TA_{271825}(n, k) = \begin{cases}1&\text{if n = 0},\\\frac {k {(-1)}^{n-1}} {n} {\binom{2n-k-1}{n-1}} &\text{if n > 0},\\\end{cases} $$$$T_{762}(n, m, k) = \frac{3 k \operatorname{TA_{271825}}{\left(m,k + n \right)} {\binom{3 k + 2 n}{n}}}{3 k + 2 n}$$