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

Terms

a(0) =1a(1) =1a(2) =3a(3) =13a(4) =47a(5) =171a(6) =651a(7) =2507a(8) =9703a(9) =37831a(10) =148393a(11) =584673a(12) =2312267a(13) =9174179a(14) =36500257a(15) =145566333a(16) =581746503a(17) =2329206823a(18) =9341025429a(19) =37516150599a(20) =150874376997a(21) =607479424817