Generating function
$$U_{1366}(x, y) = \frac{2 \sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt[4]{3} \sqrt{\frac{x \sin{\left(\frac{\operatorname{asin}{\left(3 \sqrt{3} \sqrt{y} \right)}}{3} \right)}}{\sqrt{y}}}}{2} \right)}}{3} \right)}}{3 \sqrt{x}}$$
Explicit formula
$$T_{1366}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{k + 3 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + n - 1}{m}}}{k + 4 m + 2 n}&\text{if k even} ,\ \\\frac{k {\binom{k + 3 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + n - \frac{1}{2}}{m}} {\binom{k + 6 m + 2 n - 1}{\frac{k}{2} + 3 m + n - \frac{1}{2}}}}{\left(k + 4 m + 2 n\right) {\binom{k + 4 m + 2 n - 1}{\frac{k}{2} + 2 m + n - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
Data table
1 2 22 340 6118 120060 2492028
1 6 78 1292 24150 485460 10245004
3 30 450 7980 155250 3204036 68888820
12 168 2856 54096 1096200 23258928 510221712
55 990 18810 379500 8010090 174871620 3917773860
273 6006 126126 2702700 59393334 1334665332 30558636108
1428 37128 853944 19380816 443122680 10250818032 239964130224
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