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

Terms

a(0) =1a(1) =3a(2) =13a(3) =59a(4) =271a(5) =1250a(6) =5775a(7) =26696a(8) =123423a(9) =570576a(10) =2637306a(11) =12187755a(12) =56312089a(13) =260134905a(14) =1201493926a(15) =5548533913a(16) =25619837773a(17) =118283258215a(18) =546041467522