Generating function
$$U_{1268}(x, y) = \frac{1 - \sqrt{- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}} + 1}}{2 x}$$
Explicit formula
$$T_{1268}(n, m, k) = \frac{2 k {\binom{k + 2 n - 1}{n}} {\binom{4 k + 2 m + 4 n}{m}}}{2 k + m + 2 n}$$
Data table
1 4 14 48 165 572 2002
1 8 44 208 91 3808 15504
2 24 18 1088 5814 28728 134596
5 8 76 56 3542 2024 10764
14 28 322 28 20475 1330056 792918
42 1008 13608 136416 113274 8249472 54538176
132 3696 57288 650496 604758 48820464 354666312
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