$$U_{969}(x, y) = \frac{\sqrt{4 x + 1} \left(\sqrt{4 y + 1} + 1\right)}{2}$$

$$TA_{271825}(n, k) = \begin{cases}1&\text{if n = 0},\\\frac {k {(-1)}^{n-1}} {n} {\binom{2n-k-1}{n-1}} &\text{if n > 0},\\\end{cases} $$$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{969}(n, m, k) = \operatorname{TA_{271825}}{\left(m,k \right)} \operatorname{Tsqrt}{\left(n,k \right)}$$