$$U_{871}(x, y) = \left(1 - y\right)^{5} \left(x + \frac{1}{\left(1 - y\right)^{8}}\right)$$

$$T_{871}(n, m, k) = {\binom{k}{n}} {\binom{3 k + m - 8 n - 1}{m}}$$