This sequence shows the cycle type of each finite permutation (A195663) as the index number of the corresponding partition. (When a permutation has a 3-cycle and a 2-cycle, this corresponds to the partition 3+2, etc.) Partitions can be ordered, so each partition can be denoted by its index in this order, e.g. 6 for the partition 3+2. Compare A194602.
Entries 1,2,4,6,10,14,21... ( A000041(n)-1 from n=2 ) correspond to permutations with exactly one n-cycle (and no other cycles).
Entries 1,3,7,15,30,56,101... ( A000041(2n-1) from n=1 ) correspond to permutations with exactly n 2-cycles (and no other cycles), so these are the symmetric permutations.
Entries n = 1,3,4,7,9,10,12... ( A194602(n) has an even binary digit sum ) correspond to even permutations. This goes along with the fact, that a permutation is even when its partition contains an even number of even addends.
Cf. A195663, A195664, A055089 (ordered finite permutations).
Cf. A194602 (ordered partitions interpreted as binary numbers).
Cf. A181897 (number of n-permutations with cycle type k).