Pyramid #618
Generating function
$$U_{618}(x, y) = - \frac{\sqrt{4 x + \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}}{2} - \frac{2 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3} + \frac{1}{2}$$
Explicit formula
$$T_{618}(n, m, k) = \begin{cases}0&\text{if n < k} ,\ \\- \frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\- \frac{\left(-1\right)^{m - 1} k \left(k - 2 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 3 m + 2 n - 1}{m - 1}}}{m n} \end{cases} $$
Data table
0 0 0 0 0 0 0
-1 1 -3 12 -55 273 -1428
1 -3 12 -55 273 -1428 7752
-2 1 -5 26 -14 7752 -4389
5 -35 21 -1225 714 -41895 24794
-14 126 -882 5712 -3591 223146 -138138
42 -462 3696 -26334 17787 -116886 756756
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