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

Terms

a(0) =1a(1) =3a(2) =9a(3) =34a(4) =111a(5) =351a(6) =1103a(7) =3384a(8) =10224a(9) =30536a(10) =90222a(11) =264186a(12) =767663a(13) =2215623a(14) =6356907a(15) =18143300a(16) =51540885a(17) =145801395a(18) =410888595a(19) =1153964520a(20) =3230723826a(21) =9019081038a(22) =25112021154a(23) =69750583164a(24) =193303849531a(25) =534602071341