Generating function
$$U_{1376}(x, y) = \frac{\left(y + 1\right)^{2}}{\sqrt{- 4 x \left(y + 1\right)^{2} + 1}}$$
Explicit formula
$$T_{1376}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\4^{n} {\binom{2 k + 2 n}{m}} {\binom{\frac{k}{2} + n - 1}{n}}&\text{if k even} ,\ \\\frac{{\binom{2 k + 2 n}{m}} {\binom{\frac{k}{2} + n - \frac{1}{2}}{n}} {\binom{k + 2 n - 1}{\frac{k}{2} + n - \frac{1}{2}}}}{{\binom{k - 1}{\frac{k}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
Data table
1 2 1 0 0 0 0
2 8 12 8 2 0 0
6 36 9 12 9 36 6
2 16 56 112 14 112 56
7 7 315 84 147 1764 147
252 3024 16632 5544 12474 199584 232848
924 12936 84084 336336 924924 1849848 2774772
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