Generating function
$$U_{548}(x, y) = \frac{\sqrt{3} \sqrt{y} \left(\sqrt{4 x + \frac{16 \sin^{4}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{2}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)}{4 \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Explicit formula
$$T_{548}(n, m, k) = \begin{cases}1&\text{if n==0, m==0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0, n>0} ,\ \\\frac{k {\binom{k + 3 m - 1}{m}}}{k + 2 m}&\text{if n==0, m>0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - 4 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k - 2 m + 4 n - 1}{m - 1}}}{m n}&\text{if n>0, m>0} \end{cases} $$
1 | 1 | 3 | 12 | 55 | 273 | 1428 |
1 | -3 | -3 | -1 | -42 | -198 | -1001 |
-1 | 7 | -7 | 0 | 7 | 49 | 294 |
2 | -22 | 66 | -44 | 0 | 0 | -22 |
-5 | 75 | -375 | 7 | -375 | 15 | 0 |
14 | -266 | 1862 | -5852 | 798 | -3724 | 266 |
-42 | 966 | -8694 | 3864 | -87906 | 95634 | -40572 |
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