$$U_{817}(x, y) = \frac{1}{\left(- \frac{x}{\sqrt{1 - 4 y}} + 1\right)^{2}}$$

$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$T_{817}(n, m, k) = \operatorname{TA_{984}}{\left(m,n \right)} {\binom{2 k + n - 1}{n}}$$