a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*J(k), where J = A001045.

Terms

a(0) =0a(1) =0a(2) =1a(3) =3a(4) =7a(5) =15a(6) =33a(7) =77a(8) =187a(9) =459a(10) =1121a(11) =2717a(12) =6555a(13) =15795a(14) =38081a(15) =91893a(16) =221867a(17) =535755a(18) =1293633a(19) =3123277a(20) =7540187a(21) =18203139a(22) =43945441a(23) =106092997a(24) =256131435a(25) =618357915a(26) =1492851361a(27) =3604064733a(28) =8700980827a(29) =21006018195