Pyramid #293
Generating function
$$U_{293}(x, y) = x + \frac{2 \sqrt{3} \sin{\left(\frac{\arcsin{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}}$$
Explicit formula
$$T_{293}(n, m, k) = \begin{cases}{\binom{k}{n}}&\text{if m=0} ,\ \\0&\text{if m>0 and 2*m+k-n=0 } ,\ \\\frac{\left(k - n\right) {\binom{k}{n}} {\binom{k + 3 m - n - 1}{m}}}{k + 2 m - n} \end{cases} $$
Data table
1 1 3 12 55 273 1428
1 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
Export
expand_less