Generating function
$$U_{1461}(x, y) = \frac{- 2 x \left(y + 1\right)^{2} - \sqrt{- 4 x \left(y + 1\right)^{2} + 1} + 1}{2 x^{2} \left(y + 1\right)}$$
Explicit formula
$$T_{1461}(n, m, k) = \frac{k {\binom{2 k + 2 n}{n}} {\binom{3 k + 2 n}{m}}}{k + n}$$
Data table
1 3 3 1 0 0 0
2 1 2 2 1 2 0
5 35 105 175 175 105 35
14 126 504 1176 1764 1764 1176
42 462 231 693 1386 19404 19404
132 1716 10296 37752 9438 169884 226512
429 6435 45045 195195 585585 1288287 2147145
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