a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041.

Terms

a(0) =1a(1) =3a(2) =8a(3) =19a(4) =43a(5) =93a(6) =197a(7) =409a(8) =840a(9) =1710a(10) =3462a(11) =6980a(12) =14037a(13) =28175a(14) =56485a(15) =113146a(16) =226523a(17) =453343a(18) =907071a(19) =1814632a(20) =3629891a(21) =7260574a(22) =14522150a(23) =29045555a(24) =58092685a(25) =116187328a(26) =232377092a(27) =464757194a(28) =929518106a(29) =1859040777