Pyramid #981
Generating function
$$U_{981}(x, y) = \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x \left(y + 1\right)}}{2} \right)}}{3} \right)}}{3}\right) \left(y + 1\right)^{2}$$
Explicit formula
$$T_{981}(n, m, k) = \begin{cases}{\binom{2 k + n}{m}}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{2 k + n}{m}} {\binom{- k + 3 n - 1}{n - 1}}}{n} \end{cases} $$
Data table
1 2 1 0 0 0 0
1 3 3 1 0 0 0
-2 -8 -12 -8 -2 0 0
7 35 7 7 35 7 0
-3 -18 -45 -6 -45 -18 -3
143 1001 3003 5005 5005 3003 1001
-728 -5824 -20384 -40768 -5096 -40768 -20384
Related
Export
expand_less