Generating function
$$U_{488}(x, y) = \frac{- 4 x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)} + \sqrt{3} \sqrt{y} - \sqrt{- 8 \sqrt{3} x \sqrt{y} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)} + 3 y}}{4 x^{2} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Explicit formula
$$T_{488}(n, m, k) = \frac{k {\binom{2 k + 2 n}{n}} {\binom{k + 3 m + n - 1}{m}}}{k + 2 m + n}$$
1 | 1 | 3 | 12 | 55 | 273 | 1428 |
2 | 4 | 14 | 6 | 286 | 1456 | 7752 |
5 | 15 | 6 | 275 | 1365 | 714 | 3876 |
14 | 56 | 252 | 1232 | 637 | 34272 | 189924 |
42 | 21 | 105 | 546 | 294 | 162792 | 92169 |
132 | 792 | 4356 | 24024 | 13464 | 767448 | 4441668 |
429 | 3003 | 18018 | 105105 | 612612 | 3594591 | 21273252 |
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