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

Terms

a(0) =1a(1) =1a(2) =5a(3) =7a(4) =31a(5) =49a(6) =209a(7) =351a(8) =1471a(9) =2561a(10) =10625a(11) =18943a(12) =78079a(13) =141569a(14) =580865a(15) =1066495a(16) =4361215a(17) =8085505a(18) =32978945a(19) =61616127a(20) =250806271a(21) =471556097a(22) =1916280833a(23) =3621830655a(24) =14698053631a(25) =27902803969a(26) =113104519169a(27) =215530668031a(28) =872801042431