Generating function
$$U_{372}(x, y) = \frac{\sqrt{4 x^{4} y - 16 x^{3} y + 24 x^{2} y - 16 x y + 4 y + 1} + 1}{2 x^{2} - 4 x + 2}$$
Explicit formula
$$T_{372}(n, m, k) = \begin{cases}{\binom{2 k + n - 1}{n}}&\text{if m=0} ,\ \\\frac{k {\binom{k - m - 1}{m - 1}} {\binom{2 k - 4 m + n - 1}{n}}}{m},\ \\1&\text{if n=0,m=0} \end{cases} $$
1 | 1 | -1 | 2 | -5 | 14 | -42 |
2 | -2 | 6 | -2 | 7 | -252 | 924 |
3 | 1 | -15 | 9 | -455 | 2142 | -9702 |
4 | 0 | 2 | -24 | 182 | -11424 | 6468 |
5 | 0 | -15 | 42 | -5005 | 4284 | -30723 |
6 | 0 | 6 | -504 | 1001 | -119952 | 1106028 |
7 | 0 | -1 | 42 | -15015 | 259896 | -3133746 |
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