structural:expansion, quality:correction
Simple identities

Some simple identities include

This is because dividing n elements into n − 1 sets necessarily means dividing it into one set of size 2 and n − 2 sets of size 1. Therefore we need only pick those two elements;

and

To see this, first note that there are 2 ordered pairs of complementary subsets A and B. In one case, A is empty, and in another B is empty, so 2 − 2 ordered pairs of subsets remain. Finally, since we want unordered pairs rather than ordered pairs we divide this last number by 2, giving the result above.

Another explicit expansion of the recurrence-relation gives identities in the spirit of the above example.