Pyramid #652
Generating function
$$U_{652}(x, y) = \frac{\sqrt{3} \left(\sqrt{4 x + 1} + 1\right) \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}}$$
Explicit formula
$$T_{652}(n, m, k) = \begin{cases}\frac{k {\binom{k + 3 m - 1}{m}}}{k + 2 m}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k^{2} {\binom{- k + 2 n - 1}{n - 1}} {\binom{k + 3 m - 1}{m}}}{n \left(k + 2 m\right)} \end{cases} $$
Data table
1 1 3 12 55 273 1428
1 1 3 12 55 273 1428
-1 -1 -3 -12 -55 -273 -1428
2 2 6 24 11 546 2856
-5 -5 -15 -6 -275 -1365 -714
14 14 42 168 77 3822 19992
-42 -42 -126 -504 -231 -11466 -59976
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