$$U_{1315}(x, y) = \frac{\left(1 - y\right)^{5} \left(- \frac{2 x}{1 - y} - \sqrt{- \frac{4 x}{1 - y} + 1} + 1\right)^{3}}{8 x^{6}}$$

$$T_{1315}(n, m, k) = \frac{3 k {\binom{6 k + 2 n}{n}} {\binom{k + m + n - 1}{m}}}{3 k + n}$$