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

Terms

a(0) =1a(1) =2a(2) =5a(3) =20a(4) =77a(5) =286a(6) =1065a(7) =3976a(8) =14841a(9) =55386a(10) =206701a(11) =771420a(12) =2878981a(13) =10744502a(14) =40099025a(15) =149651600a(16) =558507377a(17) =2084377906a(18) =7779004245a(19) =29031639076a(20) =108347552061a(21) =404358569166