Solution (a(n)) of the complementary equation in Comments.

Terms

a(0) =1a(1) =4a(2) =16a(3) =50a(4) =155a(5) =468a(6) =1410a(7) =4234a(8) =12709a(9) =38132a(10) =114404a(11) =343218a(12) =1029663a(13) =3088997a(14) =9267001a(15) =27801012a(16) =83403047a(17) =250209151a(18) =750627465a(19) =2251882406a(20) =6755647231a(21) =20266941705a(22) =60800825129a(23) =182402475400a(24) =547207426215