edited to reflect advice from the comments:
While working on a generalization of a tiling problem, I generated a recurrence relation to describe the total number of possible tilings. The relation contains a summation term: $a_n= 2a_{n-1}+a_{n-2}+ 4\sum\limits_{j=2}^n a_{n-j}$. I see from the answers to this question that such relations do occur and there exist several methods to solve them. Specifically, it appears that as long as the recurrence is linear, translating a given relation with a sum into one without a sum is fairly straightforward.
My question is: are there other classes of recurrence relations that also permit us to algebraically dispose of summations if they occur in the relation?