$$U_{1309}(x, y) = \frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{4 y^{3}} + \sqrt{\frac{x^{2} \left(1 - \sqrt{1 - 4 y}\right)^{6}}{16 y^{6}} + 1}$$

$$Tsqrt2(n, k) = \begin{cases}\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k even, where j =} \frac{n+k} {2},\\\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k odd, where j =} \frac{n+k+1} {2},\\\end{cases} $$$$T_{1309}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0} ,\ \\\frac{3 n \operatorname{Tsqrt_{2}}{\left(n,k \right)} {\binom{2 m + 3 n - 1}{m}}}{m + 3 n} \end{cases} $$