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
$$T_{1366}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{k + 3 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + n - 1}{m}}}{k + 4 m + 2 n}&\text{if k even} ,\ \\\frac{k {\binom{k + 3 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + n - \frac{1}{2}}{m}} {\binom{k + 6 m + 2 n - 1}{\frac{k}{2} + 3 m + n - \frac{1}{2}}}}{\left(k + 4 m + 2 n\right) {\binom{k + 4 m + 2 n - 1}{\frac{k}{2} + 2 m + n - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
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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