a(n)=sum{k=0..floor(n/2), binomial(n-k,k)*A186338(k)}.

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =15a(5) =33a(6) =95a(7) =237a(8) =667a(9) =1765a(10) =4943a(11) =13505a(12) =37967a(13) =105837a(14) =299675a(15) =847253a(16) =2417903a(17) =6909409a(18) =19866303a(19) =57253165a(20) =165728475a(21) =480938693a(22) =1400391247a(23) =4087481409a(24) =11963060527a(25) =35089773869a(26) =103157489499a(27) =303856951925a(28) =896755068783