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
$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$T_{874}(n, m, k) = \frac{k \operatorname{TA_{984}}{\left(m,k + n \right)} {\binom{k + 3 n - 1}{n}}}{k + 2 n}$$
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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