$$U_{820}(x, y) = \frac{1 - \sqrt{- 4 x y^{5} - 20 x y^{4} - 40 x y^{3} - 40 x y^{2} - 20 x y - 4 x + 1}}{2 x y^{2} + 4 x y + 2 x}$$

$$T_{820}(n, m, k) = \frac{k {\binom{k + 2 n}{n}} {\binom{3 k + 5 n}{m}}}{k + 2 n}$$