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

Terms

a(0) =1a(1) =4a(2) =15a(3) =55a(4) =200a(5) =726a(6) =2640a(7) =9636a(8) =35343a(9) =130339a(10) =483395a(11) =1802901a(12) =6760781a(13) =25482643a(14) =96506229a(15) =367077447a(16) =1401772536a(17) =5372120718a(18) =20653929804a(19) =79634421312a(20) =307826528346