a(n) = Sum_{k=0..floor(n/3)} n^k * binomial(2*n-3*k-1,n-3*k).

Terms

a(0) =1a(1) =1a(2) =3a(3) =13a(4) =51a(5) =201a(6) =834a(7) =3529a(8) =15075a(9) =65431a(10) =288278a(11) =1285263a(12) =5799470a(13) =26492103a(14) =122432628a(15) =572291385a(16) =2705760291a(17) =12937116213a(18) =62542367166a(19) =305668511259