a(n) = [x^n](1/2)*(1 + (2*x + 1)/sqrt(1 - 8*x^2*(x + 1))).

Terms

a(0) =1a(1) =1a(2) =2a(3) =6a(4) =16a(5) =48a(6) =140a(7) =424a(8) =1280a(9) =3920a(10) =12032a(11) =37184a(12) =115248a(13) =358624a(14) =1118784a(15) =3499584a(16) =10969344a(17) =34450944a(18) =108377984a(19) =341465344a(20) =1077300224a(21) =3403006464a(22) =10761447424a(23) =34065967104a(24) =107937899264a(25) =342293526016