a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,k) * binomial(n-k-1,n-3*k).

Terms

a(0) =1a(1) =0a(2) =0a(3) =3a(4) =8a(5) =15a(6) =45a(7) =147a(8) =408a(9) =1128a(10) =3325a(11) =9801a(12) =28421a(13) =82953a(14) =244293a(15) =719583a(16) =2119512a(17) =6259485a(18) =18525060a(19) =54879980a(20) =162758613a(21) =483316683a(22) =1436757839a(23) =4274856843a(24) =12730177173a(25) =37940570190a(26) =113160456435a(27) =337738702323