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

Terms

a(0) =1a(1) =3a(2) =5a(3) =10a(4) =19a(5) =35a(6) =65a(7) =120a(8) =221a(9) =407a(10) =749a(11) =1378a(12) =2535a(13) =4663a(14) =8577a(15) =15776a(16) =29017a(17) =53371a(18) =98165a(19) =180554a(20) =332091a(21) =610811a(22) =1123457a(23) =2066360a(24) =3800629a(25) =6990447a(26) =12857437a(27) =23648514a(28) =43496399a(29) =80002351