Generating function
$$U_{1364}(x, y) = \frac{3^{\frac{3}{4}} \sqrt{\frac{\sin{\left(\frac{\operatorname{asin}{\left(3 \sqrt{3} \sqrt{y} \right)}}{3} \right)}}{\sqrt{y}}}}{3 \left(1 - x\right)}$$
Explicit formula
$$T_{1364}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{\frac{k}{2} + 3 m - 1}{m}} {\binom{k + n - 1}{n}}}{k + 4 m}&\text{if k even} ,\ \\\frac{k {\binom{\frac{k}{2} + 3 m - \frac{1}{2}}{m}} {\binom{k + 6 m - 1}{\frac{k}{2} + 3 m - \frac{1}{2}}} {\binom{k + n - 1}{n}}}{\left(k + 4 m\right) {\binom{k + 4 m - 1}{\frac{k}{2} + 2 m - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
Data table
1 2 22 34 6118 12006 2492028
1 2 22 34 6118 12006 2492028
1 2 22 34 6118 12006 2492028
1 2 22 34 6118 12006 2492028
1 2 22 34 6118 12006 2492028
1 2 22 34 6118 12006 2492028
1 2 22 34 6118 12006 2492028
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