A family of marked Motzkin-like paths.

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =3a(5) =9a(6) =21a(7) =41a(8) =79a(9) =169a(10) =393a(11) =913a(12) =2051a(13) =4537a(14) =10165a(15) =23257a(16) =53759a(17) =124153a(18) =286009a(19) =660161a(20) =1531875a(21) =3571753a(22) =8348981a(23) =19539209a(24) =45792719a(25) =107546633a(26) =253153609a(27) =597034609a(28) =1410131683a(29) =3334984025