Generating function
$$U_{1264}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4} \left(- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}} + 1\right)^{3}}$$
Explicit formula
$$T_{1264}(n, m, k) = \frac{\left(2 k + 2 n\right) {\binom{3 k + n - 1}{n}} {\binom{4 k + 2 m + 4 n}{m}}}{2 k + m + 2 n}$$
Data table
1 4 14 48 165 572 2002
3 24 132 624 273 11424 46512
6 72 54 3264 17442 86184 403788
1 16 152 112 7084 4048 21528
15 3 345 3 219375 142506 849555
21 504 6804 68208 56637 4124736 27269088
28 784 12152 137984 128282 10355856 75232248
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