Pyramid #368
Generating function
$$U_{368}(x, y) = \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{3}}\right)}}{18} + \frac{1}{27 \left(1 - y\right)^{3}}} + \frac{1}{3 - 3 y} + \frac{1}{9 \left(1 - y\right)^{2} \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{3}}\right)}}{18} + \frac{1}{27 \left(1 - y\right)^{3}}}}$$
Explicit formula
$$T_{368}(n, m, k) = \begin{cases}1&\text{if n=0} ,\ \\\frac{k {\binom{k - 2 n - 1}{n - 1}} {\binom{k + m - 3 n - 1}{m}}}{n},\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + n - 1}{m - 1}}}{m}&\text{if n=0} \end{cases} $$
Data table
1 1 1 1 1 1 1
1 -2 1 0 0 0 0
-2 1 -2 2 -1 2 0
7 -56 196 -392 49 -392 196
-3 33 -165 495 -99 1386 -1386
143 -2002 13013 -52052 143143 -286286 429429
-728 12376 -99008 49504 -173264 4504864 -9009728
Export
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