Generating function
$$U_{818}(x, y) = \frac{1}{\sqrt{- \frac{4 x}{\sqrt{1 - 4 y}} + 1}}$$
Explicit formula
$$TA_{984}(n, k) = \begin{cases}{4}^{n} \binom {n+j-1} {n} &\text{if k even, where j =} \frac {k} {2},\\\frac {\binom {n+j} {n} \binom {2n+2j} {n+j}} {\binom {2j} {j}}&\text{if k odd, where j =} \frac {k-1} {2},\end{cases} $$$$T_{818}(n, m, k) = \operatorname{TA_{984}}{\left(m,n \right)} \operatorname{TA_{984}}{\left(n,k \right)}$$
1 | 0 | 0 | 0 | 0 | 0 | 0 |
2 | 4 | 12 | 40 | 140 | 504 | 1848 |
6 | 24 | 96 | 384 | 1536 | 6144 | 24576 |
20 | 120 | 600 | 2800 | 12600 | 55440 | 240240 |
70 | 560 | 3360 | 17920 | 89600 | 430080 | 2007040 |
252 | 2520 | 17640 | 105840 | 582120 | 3027024 | 15135120 |
924 | 11088 | 88704 | 591360 | 3548160 | 19869696 | 105971712 |
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