a(n) = sum{k=0..floor(n/2), C(2*n-3*k, n)*C(n-k, k)}.

Terms

a(0) =1a(1) =2a(2) =6a(3) =22a(4) =85a(5) =336a(6) =1350a(7) =5492a(8) =22554a(9) =93300a(10) =388201a(11) =1622868a(12) =6811056a(13) =28680356a(14) =121111440a(15) =512684484a(16) =2174928031a(17) =9243973062a(18) =39354962345a(19) =167799259130a(20) =716414975613