counting

1,2,2,3,4,4,4=7!/3!2! ==420 permutations
1,2,2,4,4,4,5=7!/3!2! ==420 permutations
1,2,3,4,4,4,6=7!/3!2! ==420 permutations

420 x 3 ==1,260 possible sequences of rolls

You started with the right idea, but you have to do it for all 6 digits:

1122444=7!/3!2!2! =210
1222444=7!/3!3! =140
1223444=7!/3!2! = 420
1224444 =7!/4!2! = 105
1224445 =7!/3!2! = 420
1224446 = 7!/3!2! = 420
The total =210+140+420+105+420+420==1715 possible sequences of rolls