Pyramid #476
Generating function
$$U_{476}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2} - \sqrt{4 x \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right) + \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}}}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}}$$
Explicit formula
$$T_{476}(n, m, k) = \begin{cases}0&\text{if n<k} ,\ \\- \frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\- \frac{\left(-1\right)^{m - 1} k \left(k - 3 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 3 m + 3 n - 1}{m - 1}}}{m n}&\text{if m>0 and n>0} \end{cases} $$
Data table
0 0 0 0 0 0 0
-1 2 -7 3 -143 728 -3876
1 -5 25 -13 7 -3876 21945
-2 16 -104 64 -3876 23408 -14168
5 -55 44 -3135 21175 -13915 9009
-14 196 -1862 15092 -1127 80262 -5550426
42 -714 7854 -714 584766 -4493202 33112464
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