a(n) = Sum_{k=0..floor(n/4)} binomial(n+3,4*k+3) * Catalan(k).

Terms

a(0) =1a(1) =4a(2) =10a(3) =20a(4) =36a(5) =64a(6) =120a(7) =240a(8) =497a(9) =1036a(10) =2158a(11) =4524a(12) =9625a(13) =20816a(14) =45560a(15) =100368a(16) =221915a(17) =492380a(18) =1097302a(19) =2457228a(20) =5526666a(21) =12474000a(22) =28233600a(23) =64061920a(24) =145704327a(25) =332174532a(26) =758977386a(27) =1737703780a(28) =3985847284a(29) =9157908736