$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$8$ |
$0$ |
$96$ |
$0$ |
$1280$ |
$0$ |
$17920$ |
$0$ |
$258048$ |
$0$ |
$3784704$ |
$a_2$ |
$1$ |
$4$ |
$18$ |
$88$ |
$454$ |
$2424$ |
$13236$ |
$73392$ |
$411462$ |
$2325976$ |
$13233628$ |
$75682512$ |
$434662684$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$4$ |
$8$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$18$ |
$40$ |
$96$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$88$ |
$208$ |
$512$ |
$1280$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$454$ |
$1112$ |
$2784$ |
$7040$ |
$17920$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$2424$ |
$6064$ |
$15360$ |
$39168$ |
$100352$ |
$258048$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$13236$ |
$33552$ |
$85696$ |
$219904$ |
$566272$ |
$1462272$ |
$3784704$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&4&0&6&0&11&0&10\\0&8&0&0&24&0&24&0&48&0\\3&0&11&18&0&24&0&51&0&42\\4&0&18&34&0&42&0&98&0&76\\0&24&0&0&80&0&88&0&168&0\\6&0&24&42&0&56&0&126&0&102\\0&24&0&0&88&0&104&0&192&0\\11&0&51&98&0&126&0&301&0&236\\0&48&0&0&168&0&192&0&368&0\\10&0&42&76&0&102&0&236&0&192\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&8&11&34&80&56&104&301&368&192\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.