a(n) = Sum_{k=0..n} (k+1) * 2^(n-k) * binomial(k,n-k)^2.

Terms

a(0) =1a(1) =2a(2) =7a(3) =28a(4) =89a(5) =310a(6) =1059a(7) =3552a(8) =11973a(9) =40074a(10) =133647a(11) =444708a(12) =1475441a(13) =4884574a(14) =16139387a(15) =53231976a(16) =175300509a(17) =576477618a(18) =1893341927a(19) =6211198636a(20) =20354724809a(21) =66640367494a(22) =217984687571a(23) =712460959984