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

Terms

a(0) =1a(1) =1a(2) =1a(3) =3a(4) =2a(5) =2a(6) =4a(7) =7a(8) =2a(9) =2a(10) =15a(11) =18a(12) =24a(13) =4a(14) =4a(15) =24a(16) =57a(17) =30a(18) =36a(19) =4a(20) =4a(21) =105a(22) =174a(23) =282a(24) =88a(25) =100a(26) =8a(27) =8a(28) =192a(29) =561