Pyramid #874
Generating function
$$U_{874}(x, y) = \frac{2 \sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{\frac{x}{\sqrt{1 - 4 y}}}}{2} \right)}}{3} \right)}}{3 \sqrt{\frac{x}{\sqrt{1 - 4 y}}} \sqrt{1 - 4 y}}$$
Explicit formula
$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$T_{874}(n, m, k) = \frac{k \operatorname{TA_{984}}{\left(m,k + n \right)} {\binom{k + 3 n - 1}{n}}}{k + 2 n}$$
Data table
1 2 6 20 70 252 924
1 4 16 64 256 1024 4096
3 18 90 420 1890 8316 36036
12 96 576 3072 15360 73728 344064
55 550 3850 23100 127050 660660 3303300
273 3276 26208 174720 1048320 5870592 31309824
1428 19992 179928 1319472 8576568 51459408 291603312
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