a(n) = 1 + (1200 + (634 + (225 + (85 + (15 + n)*n)*n)*n)*n)*n/720.

Terms

a(0) =1a(1) =4a(2) =13a(3) =38a(4) =99a(5) =231a(6) =490a(7) =960a(8) =1761a(9) =3058a(10) =5071a(11) =8086a(12) =12467a(13) =18669a(14) =27252a(15) =38896a(16) =54417a(17) =74784a(18) =101137a(19) =134806a(20) =177331a(21) =230483a(22) =296286a(23) =377040a(24) =475345a(25) =594126a(26) =736659a(27) =906598a(28) =1108003a(29) =1345369