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

Terms

a(0) =1a(1) =1a(2) =1a(3) =3a(4) =6a(5) =13a(6) =34a(7) =84a(8) =230a(9) =653a(10) =1893a(11) =5794a(12) =18080a(13) =58345a(14) =193761a(15) =657959a(16) =2295398a(17) =8177305a(18) =29775086a(19) =110676222a(20) =419169483a(21) =1617868052a(22) =6353518921a(23) =25376986471a(24) =103017630200a(25) =424704411564