Pyramid #351
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
$$T_{351}(n, m, k) = \begin{cases}{\binom{2 k + m - 1}{m}}&\text{if n=0} ,\ \\\frac{2 k {\binom{2 k - 2 n - 1}{n - 1}} {\binom{2 k + m - 4 n - 1}{m}}}{n}&\text{if n>0} \end{cases} $$
Data table
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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