Generating function
$$U_{1164}(x, y) = \frac{16 y^{6} \left(1 - \sqrt{- \frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{2 y^{3}} + 1}\right)^{2}}{x^{2} \left(1 - \sqrt{1 - 4 y}\right)^{6}}$$
Explicit formula
$$T_{1164}(n, m, k) = \frac{6 k n {\binom{2 k + 2 n - 1}{n}} {\binom{2 m + 3 n - 1}{m}}}{\left(2 k + n\right) \left(m + 3 n\right)}$$
Data table
nan 0 0 0 0 0 0
2 6 18 56 18 594 2002
5 3 135 55 2145 819 3094
14 126 756 3822 1764 77112 325584
42 504 378 22848 122094 603288 2826516
132 198 1782 1254 7623 4207896 217074
429 7722 81081 651222 444015 2710422 152837685
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