Generating function
$$U_{290}(x, y) = \frac{\sqrt{3} \sqrt{y} - \sqrt{- 16 x \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)} + 3 y}}{4 x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Explicit formula
$$T_{290}(n, m, k) = \frac{k {\binom{k + 2 n}{n}} {\binom{k + 3 m + 2 n - 1}{m}}}{k + 2 m + 2 n}$$
1 | 1 | 3 | 12 | 55 | 273 | 1428 |
1 | 3 | 12 | 55 | 273 | 1428 | 7752 |
2 | 1 | 5 | 26 | 14 | 7752 | 4389 |
5 | 35 | 21 | 1225 | 714 | 41895 | 24794 |
14 | 126 | 882 | 5712 | 3591 | 223146 | 138138 |
42 | 462 | 3696 | 26334 | 17787 | 116886 | 756756 |
132 | 1716 | 15444 | 12012 | 868296 | 602316 | 40756716 |
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