a(n) = Sum_{i+j+k=n, 0<=j<=i<=k<=n} (n+k)!/(i! * j! * (2*k)!).

Terms

a(0) =1a(1) =1a(2) =3a(3) =16a(4) =41a(5) =166a(6) =777a(7) =2472a(8) =9781a(9) =44506a(10) =150953a(11) =597334a(12) =2661881a(13) =9337511a(14) =37044451a(15) =162468924a(16) =582650861a(17) =2315651686a(18) =10035973071a(19) =36558456606a(20) =145470310301a(21) =624753156831