Pyramid #490
Generating function
$$U_{490}(x, y) = \frac{\sqrt{3} \left(\sqrt{3} x \sqrt{y} - \sqrt{2 \sqrt{3} x \sqrt{y} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)} + \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}} + \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}\right)}{3 \sqrt{y}}$$
Explicit formula
$$T_{490}(n, m, k) = \begin{cases}0&\text{if n<2*k} ,\ \\\frac{2 \left(-1\right)^{n} k {\binom{- 2 k + 2 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m} k {\binom{2 k - n - 1}{n}} {\binom{- k - 2 m + n - 1}{m - 1}}}{m} \end{cases} $$
Data table
0 0 0 0 0 0 0
0 0 0 0 0 0 0
1 -1 -2 -7 -3 -143 -728
-2 4 6 2 84 396 2002
5 -15 -15 -5 -21 -99 -5005
-14 56 28 112 49 2352 12012
42 -21 0 -21 -105 -5292 -2772
Related
Export
expand_less