Pyramid #545
Generating function
$$U_{545}(x, y) = \frac{- \sqrt{- \frac{32 \sqrt{3} x \sin^{3}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{\frac{3}{2}}} + \frac{16 \sin^{4}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{2}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}}{2 x}$$
Explicit formula
$$T_{545}(n, m, k) = \begin{cases}\frac{k {\binom{k + 2 n - 1}{n}}}{k + n}&\text{if m==0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - n\right) {\binom{k + 2 n - 1}{n}} {\binom{- k - 2 m + n - 1}{m - 1}}}{m \left(k + n\right)} \end{cases} $$
Data table
1 1 3 12 55 273 1428
1 0 0 0 0 0 0
2 -2 -4 -14 -6 -286 -1456
5 -1 -15 -5 -21 -99 -5005
14 -42 -42 -14 -588 -2772 -14014
42 -168 -84 -336 -147 -7056 -36036
132 -66 0 -66 -33 -16632 -8712
Export
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