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

Terms

a(0) =1a(1) =2a(2) =5a(3) =16a(4) =53a(5) =206a(6) =817a(7) =3620a(8) =16361a(9) =80218a(10) =401501a(11) =2139512a(12) =11641885a(13) =66599846a(14) =388962953a(15) =2367284236a(16) =14700573137a(17) =94523836850a(18) =619674301621