Generating function
$$U_{1365}(x, y) = \frac{1 - \sqrt{- \frac{4 \cdot 3^{\frac{3}{4}} x \sqrt{\frac{\sin{\left(\frac{\operatorname{asin}{\left(3 \sqrt{3} \sqrt{y} \right)}}{3} \right)}}{\sqrt{y}}}}{3} + 1}}{2 x}$$
Explicit formula
$$T_{1365}(n, m, k) = \begin{cases}\frac{4^{m} k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + \frac{n}{2} - 1}{m}}}{k + 4 m + n}&\text{if k even} ,\ \\1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + \frac{n}{2} - \frac{1}{2}}{m}} {\binom{k + 6 m + n - 1}{\frac{k}{2} + 3 m + \frac{n}{2} - \frac{1}{2}}}}{\left(k + 4 m + n\right) {\binom{k + 4 m + n - 1}{\frac{k}{2} + 2 m + \frac{n}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
1 | 2 | 22 | 340 | 6118 | 120060 | 2492028 |
1 | 3.5 | 40.8571428571429 | 646 | 11765.3846153846 | 232616.25 | 4852896.63157895 |
2 | 12 | 156 | 2584 | 48300 | 970920 | 20490008 |
5 | 36 | 487.5 | 8221.81818181818 | 155250 | 3141211.76470588 | 66592526 |
14 | 140 | 2100 | 37240 | 724500 | 14952168 | 321481160 |
42 | 462 | 7140 | 128478 | 2521260 | 52332588 | 1129776648 |
132 | 1848 | 31416 | 595056 | 12058200 | 255848208 | 5612438832 |
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