Generating function
$$U_{1287}(x, y) = \frac{8 y^{6} \left(1 - \sqrt{- \frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{2 y^{3}} + 1}\right)^{3}}{x^{3} \left(1 - \sqrt{1 - 4 y}\right)^{6}}$$
Explicit formula
$$T_{1287}(n, m, k) = \frac{9 k \left(k + n\right) {\binom{3 k + 2 n - 1}{n}} {\binom{3 k + 2 m + 3 n - 1}{m}}}{\left(3 k + n\right) \left(3 k + m + 3 n\right)}$$
Data table
1 3 9 28 9 297 1001
3 18 81 33 1287 4914 18564
9 81 486 2457 1134 49572 209304
28 336 252 15232 81396 402192 1884344
9 135 1215 855 51975 286902 148005
297 5346 56133 450846 307395 1876446 105810705
1001 21021 252252 2277275 17216199 115216101 705520816
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