Generating function
$$U_{1460}(x, y) = \frac{\left(1 - y\right)^{3} \left(- \frac{2 x}{\left(1 - y\right)^{3}} - \sqrt{- \frac{4 x}{\left(1 - y\right)^{3}} + 1} + 1\right)}{2 x^{2}}$$
Explicit formula
$$T_{1460}(n, m, k) = \frac{k {\binom{2 k + 2 n}{n}} {\binom{3 k + m + 3 n - 1}{m}}}{k + n}$$
Data table
1 3 6 1 15 21 28
2 12 42 112 252 504 924
5 45 225 825 2475 6435 15015
14 168 1092 5096 1911 61152 173264
42 63 504 2856 12852 488376 162792
132 2376 22572 15048 79002 3476088 13325004
429 9009 99099 759759 4558554 2279277 9876867
Export
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