a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A000108(k+1).

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =12a(5) =24a(6) =55a(7) =119a(8) =272a(9) =612a(10) =1411a(11) =3247a(12) =7565a(13) =17667a(14) =41561a(15) =98099a(16) =232696a(17) =553784a(18) =1322813a(19) =3169065a(20) =7614583a(21) =18342921a(22) =44294991a(23) =107200829a(24) =259983346a(25) =631718606a(26) =1537737567a(27) =3749440151a(28) =9156561590a(29) =22394270034