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

Terms

a(0) =1a(1) =1a(2) =3a(3) =13a(4) =43a(5) =171a(6) =711a(7) =2913a(8) =12363a(9) =53203a(10) =230593a(11) =1010703a(12) =4463119a(13) =19827679a(14) =88594299a(15) =397741893a(16) =1793063883a(17) =8113429419a(18) =36832823289a(19) =167701920759a(20) =765577205433