Generating function
$$U_{1488}(x, y) = \frac{\left(\frac{x}{\left(1 - 4 y\right)^{\frac{3}{2}}} + 1\right)^{2}}{\sqrt{1 - 4 y}}$$
Explicit formula
$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$T_{1488}(n, m, k) = \operatorname{TA_{984}}{\left(m,k + 3 n \right)} {\binom{2 k}{n}}$$
Data table
1 2 6 20 70 252 924
2 16 96 512 2560 12288 57344
1 14 126 924 6006 36036 204204
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
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