Generating function
$$U_{1387}(x, y) = \frac{1}{\left(1 - y\right)^{2} \sqrt{- \frac{4 x}{\left(1 - y\right)^{2}} + 1}}$$
Explicit formula
$$T_{1387}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\4^{n} {\binom{\frac{k}{2} + n - 1}{n}} {\binom{2 k + m + 2 n - 1}{m}}&\text{if k even} ,\ \\\frac{{\binom{\frac{k}{2} + n - \frac{1}{2}}{n}} {\binom{k + 2 n - 1}{\frac{k}{2} + n - \frac{1}{2}}} {\binom{2 k + m + 2 n - 1}{m}}}{{\binom{k - 1}{\frac{k}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
Data table
1 2 3 4 5 6 7
2 8 2 4 7 112 168
6 36 126 336 756 1512 2772
2 16 72 24 66 1584 3432
7 7 385 154 5005 14014 35035
252 3024 19656 91728 34398 1100736 3118752
924 12936 9702 51744 219912 7916832 25069968
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