Pyramid #614
Generating function
$$U_{614}(x, y) = - \frac{\sqrt{4 x \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right) + \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}}{2} - \frac{2 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3} + \frac{1}{2}$$
Explicit formula
$$T_{614}(n, m, k) = \begin{cases}0&\text{if n<k} ,\ \\- \frac{\left(-1\right)^{m - 1} k \left(k - n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 3 m + n - 1}{m - 1}}}{m n}&\text{if m>0} ,\ \\- \frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m = 0} \end{cases} $$
Data table
0 0 0 0 0 0 0
-1 0 0 0 0 0 0
1 -1 3 -12 55 -273 1428
-2 4 -14 6 -286 1456 -7752
5 -15 6 -275 1365 -714 3876
-14 56 -252 1232 -637 34272 -189924
42 -21 105 -546 294 -162792 92169
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