a(n,m) tabl head (triangle) for  A049029  S2(5)

 
    n\m           1             2            3            4           5        6        7        8     9   10 ...

 
    1             1             0            0            0           0        0        0        0     0    0
    
    2             5             1            0            0           0        0        0        0     0    0
    
    3            45            15            1            0           0        0        0        0     0    0
    
    4           585           255           30            1           0        0        0        0     0    0
    
    5          9945          5175          825           50           1        0        0        0     0    0
    
    6        208845        123795        24150         2025          75        1        0        0     0    0
    
    7       5221125       3427515       775845        80850        4200      105        1        0     0    0
    
    8     151412625     108046575     27478710      3363045      219450     7770      140        1     0    0
    
    9    4996616625    3824996175   1069801425    148548330    11482695   514710    13230      180     1    0
   
   10  184874815125  150346471275  45547251750   7011734625   619338825 33100515  1083600    21150   225    1

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 The row sums give  A049378: [1,6,61,871,15996,358891,9509641,290528316,10051973371,388433817091,...]. 
 
 The alternating row sums give A145561:  [1,4,31,359,5546,107249,2492701,67693534,2103854581,73651161959,...].

 The first columns give A007696 (4-factorials), A145562, A145563,...

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