Pyramid #392
Generating function
$$U_{392}(x, y) = \frac{\left(y + 1\right)^{4}}{9 \sqrt[3]{\frac{x \left(y + 1\right)^{2}}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{4}\right)} \left(y + 1\right)^{2}}{18} + \frac{\left(y + 1\right)^{6}}{27}}} + \frac{\left(y + 1\right)^{2}}{3} + \sqrt[3]{\frac{x \left(y + 1\right)^{2}}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{4}\right)} \left(y + 1\right)^{2}}{18} + \frac{\left(y + 1\right)^{6}}{27}}$$
Explicit formula
$$T_{392}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{2 k {\binom{2 k - 2 n - 1}{m - 1}}}{m}&\text{if m>0} ,\ \\\frac{k {\binom{2 k - 4 n}{m}} {\binom{k - 2 n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$
Data table
1 2 1 0 0 0 0
1 2 -1 2/3 -1/2 2/5 -1/3
-2 2 -3 4 -5 6 -7
7 2 -5 1 -35/2 28 -42
-3 2 -7 56/3 -42 84 -154
143 2 -9 3 -165/2 198 -429
-728 2 -11 44 -143 2002/5 -1001
Export
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