Generating function
$$U_{1302}(x, y) = \frac{\left(- 2 x \left(y + 1\right)^{3} - \sqrt{- 4 x \left(y + 1\right)^{3} + 1} + 1\right)^{2}}{4 x^{4} \left(y + 1\right)^{9}}$$
Explicit formula
$$T_{1302}(n, m, k) = \frac{2 k {\binom{3 k + 3 n}{m}} {\binom{4 k + 2 n}{n}}}{2 k + n}$$
Data table
1 3 3 1 0 0 0
4 24 6 8 6 24 4
14 126 504 1176 1764 1764 1176
48 576 3168 1056 2376 38016 44352
165 2475 17325 75075 225225 495495 825825
572 10296 87516 466752 175032 4900896 10618608
2002 42042 42042 266266 1198197 40738698 108636528
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