a(n) is the sum over all proper integer partitions of n of the previous terms.

Terms

a(0) =1a(1) =2a(2) =6a(3) =19a(4) =51a(5) =148a(6) =395a(7) =1095a(8) =2945a(9) =8020a(10) =21597a(11) =58518a(12) =157746a(13) =426250a(14) =1149832a(15) =3104236a(16) =8375167a(17) =22603530a(18) =60988687a(19) =164579663a(20) =444082316a(21) =1198312390a(22) =3233419264a(23) =8724918311a(24) =23542640336a(25) =63526028693a(26) =171413973501a(27) =462531951559