a(n) = Sum_{k=0..n} C(n,k)*((-1)^n*(C(k,n-k)-C(k,n-k-1))+C(n-k,k+1)).

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =19a(5) =39a(6) =141a(7) =321a(8) =1107a(9) =2675a(10) =8953a(11) =22483a(12) =73789a(13) =190345a(14) =616227a(15) =1621413a(16) =5196627a(17) =13882947a(18) =44152809a(19) =119385663a(20) =377379369a(21) =1030434069a(22) =3241135527a(23) =8921880135a(24) =27948336381a(25) =77459553549a(26) =241813226151a(27) =674100041501