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

Terms

a(0) =1a(1) =1a(2) =2a(3) =6a(4) =15a(5) =37a(6) =98a(7) =262a(8) =699a(9) =1883a(10) =5110a(11) =13918a(12) =38045a(13) =104355a(14) =287028a(15) =791320a(16) =2186209a(17) =6051113a(18) =16776022a(19) =46577806a(20) =129491865a(21) =360432855a(22) =1004332322a(23) =2801307498a(24) =7820572153a(25) =21851390549a(26) =61101872126a(27) =170977916730a(28) =478755116117