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

Terms

a(0) =1a(1) =6a(2) =25a(3) =120a(4) =581a(5) =2877a(6) =14421a(7) =72996a(8) =372229a(9) =1909336a(10) =9840909a(11) =50923041a(12) =264391973a(13) =1376654747a(14) =7185811685a(15) =37589283916a(16) =197005160825