Pyramid #364
Generating function
$$U_{364}(x, y) = \frac{\left(y + 1\right)^{4}}{9 \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{6}\right)}}{18} + \frac{\left(y + 1\right)^{6}}{27}}} + \frac{\left(y + 1\right)^{2}}{3} + \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{6}\right)}}{18} + \frac{\left(y + 1\right)^{6}}{27}}$$
Explicit formula
$$T_{364}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{2 k {\binom{2 k - 2 n - 1}{m - 1}}}{m}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 3 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{k {\binom{2 k - 6 n}{m}} {\binom{k - 2 n - 1}{n - 1}}}{n}&\text{if n>0,m>0} \end{cases} $$
Data table
1 2 1 0 0 0 0
1 -4 1 -2 35 -56 84
-2 2 -11 44 -143 4004 -1001
7 -112 952 -5712 27132 -108528 379848
-3 66 -759 6072 -3795 19734 -88803
143 -4004 58058 -58058 4499495 -28796768 158382224
-728 24752 -43316 519792 -4808076 365413776 -2375189544
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