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

Terms

a(0) =1a(1) =4a(2) =18a(3) =83a(4) =385a(5) =1788a(6) =8302a(7) =38523a(8) =178624a(9) =827665a(10) =3832613a(11) =17737631a(12) =82052146a(13) =379409431a(14) =1753785816a(15) =8104345924a(16) =37441373034a(17) =172939582903a(18) =798656132496