Generating function
$$U_{1376}(x, y) = \frac{\left(y + 1\right)^{2}}{\sqrt{- 4 x \left(y + 1\right)^{2} + 1}}$$
Explicit formula
$$T_{1376}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\4^{n} {\binom{2 k + 2 n}{m}} {\binom{\frac{k}{2} + n - 1}{n}}&\text{if k even} ,\ \\\frac{{\binom{2 k + 2 n}{m}} {\binom{\frac{k}{2} + n - \frac{1}{2}}{n}} {\binom{k + 2 n - 1}{\frac{k}{2} + n - \frac{1}{2}}}}{{\binom{k - 1}{\frac{k}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
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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