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

Terms

a(0) =1a(1) =2a(2) =3a(3) =6a(4) =12a(5) =26a(6) =62a(7) =148a(8) =396a(9) =1044a(10) =3024a(11) =8784a(12) =26928a(13) =85824a(14) =274320a(15) =954720a(16) =3149280a(17) =11910240a(18) =40253760a(19) =164643840a(20) =567181440a(21) =2497703040a(22) =8736698880a(23) =41250263040a(24) =146090649600a(25) =736680268800