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

Terms

a(0) =1a(1) =2a(2) =8a(3) =22a(4) =68a(5) =198a(6) =586a(7) =1718a(8) =5047a(9) =14808a(10) =43470a(11) =127636a(12) =374957a(13) =1102078a(14) =3241082a(15) =9537070a(16) =28079357a(17) =82718212a(18) =243809138a(19) =718994032a(20) =2121378272a(21) =6262089436a(22) =18493519148a(23) =54639914652a(24) =161503493023a(25) =477558890378