Generating function
$$U_{1318}(x, y) = \frac{4 y^{5} \left(- \frac{x \left(1 - \sqrt{1 - 4 y}\right)}{y} - \sqrt{- \frac{2 x \left(1 - \sqrt{1 - 4 y}\right)}{y} + 1} + 1\right)^{3}}{x^{6} \left(1 - \sqrt{1 - 4 y}\right)^{5}}$$
Explicit formula
$$T_{1318}(n, m, k) = \frac{3 k \left(k + n\right) {\binom{6 k + 2 n}{n}} {\binom{k + 2 m + n - 1}{m}}}{\left(3 k + n\right) \left(k + m + n\right)}$$
Data table
1 1 2 5 14 42 132
6 12 3 84 252 792 2574
27 81 243 756 243 8019 27027
11 44 154 528 1815 6292 22022
429 2145 858 32175 117975 429429 156156
1638 9828 44226 18018 702702 2683044 10135944
6188 43316 21658 952952 3941756 15767024 61855248
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