Generating function
$$U_{351}(x, y) = \frac{\frac{2 x}{3} + \frac{1}{9 \left(1 - y\right)^{4}}}{\sqrt[3]{\frac{x^{2} \left(1 - y\right)^{2}}{2} + \frac{\sqrt{3} x \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{4}}\right)} \left(1 - y\right)^{2}}{18} + \frac{x}{3 \left(1 - y\right)^{2}} + \frac{1}{27 \left(1 - y\right)^{6}}}} + \sqrt[3]{\frac{x^{2} \left(1 - y\right)^{2}}{2} + \frac{\sqrt{3} x \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{4}}\right)} \left(1 - y\right)^{2}}{18} + \frac{x}{3 \left(1 - y\right)^{2}} + \frac{1}{27 \left(1 - y\right)^{6}}} + \frac{1}{3 \left(1 - y\right)^{2}}$$
Explicit formula
$$T_{351}(n, m, k) = \begin{cases}{\binom{2 k + m - 1}{m}}&\text{if n=0} ,\ \\\frac{2 k {\binom{2 k - 2 n - 1}{n - 1}} {\binom{2 k + m - 4 n - 1}{m}}}{n}&\text{if n>0} \end{cases} $$
1 | 2 | 3 | 4 | 5 | 6 | 7 |
2 | -4 | 2 | 0 | 0 | 0 | 0 |
-3 | 18 | -45 | 6 | -45 | 18 | -3 |
1 | -1 | 45 | -12 | 21 | -252 | 21 |
-42 | 588 | -3822 | 15288 | -42042 | 84084 | -126126 |
198 | -3564 | 30294 | -161568 | 60588 | -1696464 | 3675672 |
-1001 | 22022 | -231231 | 154154 | -7322315 | 26360334 | -74687613 |
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