Generating function
$$U_{1358}(x, y) = \frac{2 \sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{x} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{4 y^{2}} \right)}}{3} \right)}}{3 \sqrt{x}}$$
Explicit formula
$$T_{1358}(n, m, k) = \frac{k {\binom{k + 3 n - 1}{n}} {\binom{2 k + 2 m + 4 n}{m}}}{k + m + 2 n}$$
Data table
1 2 5 14 42 132 429
1 6 27 11 429 1638 6188
3 3 195 105 51 23256 101745
12 168 1428 9576 5586 297528 148764
55 99 10395 8349 56925 34749 19594575
273 6006 75075 702702 5486481 37795758 237573336
1428 37128 538356 575484 50642592 388804416 2697330636
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