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) =16a(5) =50a(6) =141a(7) =469a(8) =1744a(9) =6321a(10) =21990a(11) =76395a(12) =270349a(13) =966225a(14) =3450531a(15) =12303893a(16) =43955728a(17) =157509760a(18) =565530783a(19) =2032581050a(20) =7312257366a(21) =26336300266a(22) =94962280677a(23) =342740016067