Generating function
$$U_{1158}(x, y) = \frac{y^{6} \left(4 - 4 \sqrt{- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{2 y^{6}} + 1}\right)}{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}$$
Explicit formula
$$T_{1158}(n, m, k) = \frac{3 k n {\binom{2 m + 6 n}{m}} {\binom{k + 2 n - 1}{n}}}{\left(k + n\right) \left(m + 3 n\right)}$$
Data table
nan 0 0 0 0 0 0
1 6 27 11 429 1638 6188
2 24 18 1088 5814 28728 134596
5 9 945 759 5175 3159 1781325
14 336 4536 45472 37758 2749824 18179392
42 126 2079 2499 244755 20727252 15737358
132 4752 92664 129888 14661108 141604848 1214879688
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