Pyramid #875
Generating function
$$U_{875}(x, y) = \frac{2 \sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{\frac{x}{\sqrt{1 - 4 y}}}}{2} \right)}}{3} \right)}}{3 \sqrt{\frac{x}{\sqrt{1 - 4 y}}} \left(1 - 4 y\right)}$$
Explicit formula
$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$T_{875}(n, m, k) = \frac{k \operatorname{TA_{984}}{\left(m,2 k + n \right)} {\binom{k + 3 n - 1}{n}}}{k + 2 n}$$
Data table
1 4 16 64 256 1024 4096
1 6 30 140 630 2772 12012
3 24 144 768 3840 18432 86016
12 120 840 5040 27720 144144 720720
55 660 5280 35200 211200 1182720 6307840
273 3822 34398 252252 1639638 9837828 55747692
1428 22848 228480 1827840 12794880 81887232 491323392
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