a(n) = [x^n] (3*x + 1)*(1 - sqrt(1 - 4*x))/(2*x).

Terms

a(0) =1a(1) =4a(2) =5a(3) =11a(4) =29a(5) =84a(6) =258a(7) =825a(8) =2717a(9) =9152a(10) =31382a(11) =109174a(12) =384370a(13) =1366936a(14) =4903140a(15) =17718165a(16) =64442205a(17) =235717800a(18) =866573070a(19) =3200179290a(20) =11865909990a(21) =44158628280a(22) =164881364700a(23) =617507304570