Generating function
$$U_{1426}(x, y) = \frac{\left(\frac{x}{\left(1 - y\right)^{3}} + 1\right)^{2}}{\left(1 - y\right)^{2}}$$
Explicit formula
$$T_{1426}(n, m, k) = {\binom{2 k}{n}} {\binom{2 k + m + 3 n - 1}{m}}$$
Data table
1 2 3 4 5 6 7
2 1 3 7 14 252 42
1 8 36 12 33 792 1716
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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