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
$$T_{614}(n, m, k) = \begin{cases}0&\text{if n<k} ,\ \\- \frac{\left(-1\right)^{m - 1} k \left(k - n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 3 m + n - 1}{m - 1}}}{m n}&\text{if m>0} ,\ \\- \frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m = 0} \end{cases} $$
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 |
Related
Export
Export as LaTeX
Export as Maxima
Confirm deletion
Are you sure, you want to delete Pyramid #614?