Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Terms

a(0) =1a(1) =2a(2) =1a(3) =3a(4) =2a(5) =1a(6) =8a(7) =7a(8) =2a(9) =1a(10) =15a(11) =18a(12) =12a(13) =2a(14) =1a(15) =48a(16) =57a(17) =30a(18) =18a(19) =2a(20) =1a(21) =105a(22) =174a(23) =141a(24) =44a(25) =25a(26) =2a(27) =1a(28) =384a(29) =561