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

Terms

a(0) =1a(1) =6a(2) =21a(3) =76a(4) =285a(5) =1066a(6) =3977a(7) =14840a(8) =55385a(9) =206702a(10) =771421a(11) =2878980a(12) =10744501a(13) =40099026a(14) =149651601a(15) =558507376a(16) =2084377905a(17) =7779004246a(18) =29031639077a(19) =108347552060a(20) =404358569165