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
$$T_{1365}(n, m, k) = \begin{cases}\frac{4^{m} k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + \frac{n}{2} - 1}{m}}}{k + 4 m + n}&\text{if k even} ,\ \\1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 3 m + \frac{n}{2} - \frac{1}{2}}{m}} {\binom{k + 6 m + n - 1}{\frac{k}{2} + 3 m + \frac{n}{2} - \frac{1}{2}}}}{\left(k + 4 m + n\right) {\binom{k + 4 m + n - 1}{\frac{k}{2} + 2 m + \frac{n}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
Data table
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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