Pyramid #385
Generating function
$$U_{385}(x, y) = \sqrt[3]{\frac{x}{2 - 2 y} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{2}}\right)}}{18 - 18 y} + \frac{1}{27 \left(1 - y\right)^{3}}} + \frac{1}{3 - 3 y} + \frac{1}{9 \left(1 - y\right)^{2} \sqrt[3]{\frac{x}{2 - 2 y} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{2}}\right)}}{18 - 18 y} + \frac{1}{27 \left(1 - y\right)^{3}}}}$$
Explicit formula
$$T_{385}(n, m, k) = \begin{cases}1&\text{if m=0,n=0} ,\ \\\frac{k {\binom{k - 2 n - 1}{n - 1}} {\binom{k + m - 2 n - 1}{m}}}{n}&\text{if n>0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + n - 1}{m - 1}}}{m}&\text{if n=0} \end{cases} $$
Data table
1 1 1 1 1 1 1
1 -1 0 0 0 0 0
-2 6 -6 2 0 0 0
7 -35 7 -7 35 -7 0
-3 21 -63 105 -105 63 -21
143 -1287 5148 -12012 18018 -18018 12012
-728 8008 -4004 12012 -24024 336336 -336336
Export
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