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

Terms

a(0) =1a(1) =1a(2) =3a(3) =1a(4) =6a(5) =3a(6) =1a(7) =9a(8) =15a(9) =3a(10) =1a(11) =12a(12) =36a(13) =24a(14) =3a(15) =1a(16) =15a(17) =66a(18) =90a(19) =33a(20) =3a(21) =1a(22) =18a(23) =105a(24) =228a(25) =171a(26) =42a(27) =3a(28) =1a(29) =21