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

Terms

a(0) =1a(1) =3a(2) =6a(3) =16a(4) =45a(5) =111a(6) =253a(7) =561a(8) =1221a(9) =2603a(10) =5448a(11) =11238a(12) =22906a(13) =46206a(14) =92361a(15) =183151a(16) =360636a(17) =705672a(18) =1373067a(19) =2658129a(20) =5122275a(21) =9829497a(22) =18790449a(23) =35794569a(24) =67966000a(25) =128667132a(26) =242906916a(27) =457397132a(28) =859220337a(29) =1610430267