a(n) = Sum_{k=0..n} 2^k * q(k), where q(k) is A000009 (partitions into distinct parts).

Terms

a(0) =1a(1) =3a(2) =7a(3) =23a(4) =55a(5) =151a(6) =407a(7) =1047a(8) =2583a(9) =6679a(10) =16919a(11) =41495a(12) =102935a(13) =250391a(14) =610839a(15) =1495575a(16) =3592727a(17) =8573463a(18) =20632087a(19) =48943639a(20) =116052503a(21) =275436055a(22) =648729111a(23) =1521144343a(24) =3567964695a(25) =8332694039a(26) =19405656599a(27) =45175460375a(28) =104768131607a(29) =242207085079