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Hi.

n = 1: 1
n = 2: 1
n = 3: 2, 1
n = 4: 6, 6, 5, 1
n = 5: 24, 36, 46, 40, 24, 9, 1
n = 6: 120, 240, 390, 480, 514, 416, 301, 160, 64, 14, 1
n = 7: 720, 1800, 3480, 5250, 7028, 8056, 8252, 7426, 5979, 4208, 2542, 1295, 504, 139, 20, 1
n = 8: 5040, 15120, 33600, 58800, 91014, 124250, 155994, 177220, 186810, 181076, 163149, 134665, 102745, 71070, 44605, 24550, 11712, 4543, 1344, 265, 27, 1
n = 9: 40320, 141120, 352800, 695520, 1204056, 1855728, 2640832, 3473156, 4277156, 4942428, 5395818, 5561296, 5433412, 5021790, 4391304, 3625896, 2820686, 2056845, 1398299, 879339, 504762, 260613, 117748, 45178, 13845, 3156, 461, 35, 1
n = 10: 362880, 1451520, 4021920, 8769600, 16664760, 28264320, 44216040, 64324680, 88189476, 114342744, 141184014, 166279080, 187614312, 202901634, 210825718, 210403826, 201934358, 186191430, 164980407, 140216446, 114231817, 88934355, 66047166, 46576620, 31071602, 19460271, 11365652, 6112650, 2987358, 1298181, 488878, 153094, 37692, 6705, 749, 44, 1
...

[update]: remark: for readability I transposed the original table. But I did not adapt the use of "columns" and "rows" in the surrounding text, so that must be translated in mind (Gottfried Helms)

n=1 n=2 n=3 n=4 n=5  n=6  n=7    n=8      n=9       n=10   ...
-------------------------------------------------------------- 
  1   1   2   6  24  120  720   5040    40320     362880   ...
          1   6  36  240 1800  15120   141120    1451520 
              5  46  390 3480  33600   352800    4021920 
              1  40  480 5250  58800   695520    8769600 
                 24  514 7028  91014  1204056   16664760 
                  9  416 8056 124250  1855728   28264320 
                  1  301 8252 155994  2640832   44216040 
                     160 7426 177220  3473156   64324680 
                      64 5979 186810  4277156   88189476 
                      14 4208 181076  4942428  114342744 
                       1 2542 163149  5395818  141184014 
                         1295 134665  5561296  166279080 
                          504 102745  5433412  187614312 
                          139  71070  5021790  202901634 
                           20  44605  4391304  210825718 
                            1  24550  3625896  210403826 
                               11712  2820686  201934358 
                                4543  2056845  186191430 
                                1344  1398299  164980407 
                                 265   879339  140216446 
                                  27   504762  114231817 
                                   1   260613   88934355 
                                       117748   66047166 
                                        45178   46576620 
                                        13845   31071602 
                                         3156   19460271 
                                          461   11365652 
                                           35    6112650 
                                            1    2987358 
                                                 1298181 
                                                  488878 
                                                  153094 
                                                   37692 
                                                    6705 
                                                     749 
                                                      44 
                                                       1 

$$a_2 = \frac{1}{2L - 2}$$ $$a_3 = \frac{L + 2}{6L^3 - 6L^2 - 6L + 6}$$ $$a_4 = \frac{L^3 + 5L^2 + 6L + 6}{24L^6 - 24L^5 - 24L^4 + 24L^2 + 24L - 24}$$ $$a_5 = \frac{L^6 + 9L^5 + 24L^4 + 40L^3 + 46L^2 + 35L + 24}{120L^{10} - 120L^9 - 120L^8 + 240L^5 - 120L^2 - 120L + 120}$$ ...

$$a_2 = \frac{1}{2L - 2}$$ $$a_3 = \frac{L + 2}{6L^3 - 6L^2 - 6L + 6}$$ $$a_4 = \frac{L^3 + 5L^2 + 6L + 6}{24L^6 - 24L^5 - 24L^4 + 24L^2 + 24L - 24}$$ $$a_5 = \frac{L^6 + 9L^5 + 24L^4 + 40L^3 + 46L^2 + 36L + 24}{120L^{10} - 120L^9 - 120L^8 + 240L^5 - 120L^2 - 120L + 120}$$ ...