a(n) = Sum_{k=0..floor(n/4)} (5*k+1) * binomial(2*n-3*k+1,n-4*k)/(2*n-3*k+1).

Terms

a(0) =1a(1) =1a(2) =2a(3) =5a(4) =15a(5) =48a(6) =159a(7) =539a(8) =1860a(9) =6511a(10) =23061a(11) =82482a(12) =297467a(13) =1080451a(14) =3948656a(15) =14509083a(16) =53567875a(17) =198616636a(18) =739234857a(19) =2760847225a(20) =10343227960a(21) =38860033811a(22) =146379004723a(23) =552702113268