a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 2^k.

Terms

a(0) =1a(1) =0a(2) =2a(3) =2a(4) =6a(5) =14a(6) =38a(7) =110a(8) =342a(9) =1134a(10) =3990a(11) =14830a(12) =58006a(13) =237998a(14) =1021462a(15) =4574318a(16) =21325462a(17) =103287598a(18) =518768406a(19) =2697426926a(20) =14498316182a(21) =80440333998a(22) =460112203798