Pyramid #398
Generating function
$$U_{398}(x, y) = \sqrt[3]{\frac{x}{2 \sqrt{1 - 4 y}} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{1 - 4 y}\right)}}{18 \sqrt{1 - 4 y}} + \frac{1}{27 \left(1 - 4 y\right)^{\frac{3}{2}}}} + \frac{1}{\left(9 - 36 y\right) \sqrt[3]{\frac{x}{2 \sqrt{1 - 4 y}} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{1 - 4 y}\right)}}{18 \sqrt{1 - 4 y}} + \frac{1}{27 \left(1 - 4 y\right)^{\frac{3}{2}}}}} + \frac{1}{3 \sqrt{1 - 4 y}}$$
Explicit formula
$$T_{398}(n, m, k) = \begin{cases}1&\text{if m=0 and n = 0} ,\ \\\frac{4^{m} k {\binom{k - 2 n - 1}{n - 1}} {\binom{\frac{k}{2} + m - n - 1}{m}}}{n}&\text{if n>0} ,\ \\\frac{\left(-1\right)^{m - 1} \cdot 4^{m} k {\binom{- \frac{k}{2} + \frac{n}{2} - 1}{m - 1}}}{2 m}&\text{if n=0} \end{cases} $$
Data table
1 2 4 32/3 32 512/5 1024/3
1 0 0 0 0 0 0
-2 8 0 0 0 0 0
7 -56 112 0 0 0 0
-3 36 -144 192 0 0 0
143 -2288 13728 -36608 36608 0 0
-728 1456 -11648 46592 -93184 745472 0
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