Triangle begins:
n\k 0 1 2 3 4 5 6 7 8
+
0  1
1  1 1
2  1 1 1
3  1 1 2 4
4  1 1 3 6 16
5  1 1 4 12 37 140
6  1 1 6 24 105 454 1987
7  1 1 10 40 250 1566 9856 62266
8  1 1 15 80 726 5670 47394 406168 3899340
A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways:
+++++ +++++ +++++ +++++
                   
+++++ +++++ +++++ +++++
                
+++++ + +++ ++ ++ +++ +
               
+++ + +++++ +++++ +++++
              
+ +++ + +++ + +++ + +++
               
+++++ +++++ +++++ +++++
.
+++++ +++++ +++++ +++++
                 
++ ++ +++++ +++++ + +++
                
+++++ + +++ +++++ +++++
                  
+++++ +++++ +++++ +++++
              
+ +++ ++ ++ + + + + +++
              
+++++ +++++ +++++ +++++
.
+++++ +++++ +++++ +++++
                  
+++ + +++++ +++++ +++++
                
+++++ ++ ++ +++++ +++ +
              
+++++ +++++ + + + + +++
              
+ +++ ++ ++ +++++ +++++
                 
+++++ +++++ +++++ +++++
.
+++++
   
++ ++
   
+++++
    
+++++
   
++ ++
   
+++++
The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.
