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

Terms

a(0) =1a(1) =1a(2) =5a(3) =23a(4) =111a(5) =562a(6) =2952a(7) =15948a(8) =88076a(9) =495077a(10) =2823293a(11) =16295020a(12) =95007654a(13) =558765743a(14) =3310999269a(15) =19748462718a(16) =118471172054a(17) =714355994997