Pyramid #548
Generating function
$$U_{548}(x, y) = \frac{\sqrt{3} \sqrt{y} \left(\sqrt{4 x + \frac{16 \sin^{4}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{2}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)}{4 \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Explicit formula
$$T_{548}(n, m, k) = \begin{cases}1&\text{if n==0, m==0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0, n>0} ,\ \\\frac{k {\binom{k + 3 m - 1}{m}}}{k + 2 m}&\text{if n==0, m>0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - 4 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k - 2 m + 4 n - 1}{m - 1}}}{m n}&\text{if n>0, m>0} \end{cases} $$
Data table
1 1 3 12 55 273 1428
1 -3 -3 -1 -42 -198 -1001
-1 7 -7 0 7 49 294
2 -22 66 -44 0 0 -22
-5 75 -375 7 -375 15 0
14 -266 1862 -5852 798 -3724 266
-42 966 -8694 3864 -87906 95634 -40572
Export
expand_less