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

Terms

a(0) =0a(1) =2a(2) =3a(3) =10a(4) =30a(5) =101a(6) =350a(7) =1250a(8) =4548a(9) =16782a(10) =62579a(11) =235273a(12) =890331a(13) =3387204a(14) =12943353a(15) =49643762a(16) =191010623a(17) =736946570a(18) =2850013623a(19) =11044973890a(20) =42882986660a(21) =166770990377a(22) =649526893537