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) =7a(5) =3a(6) =1a(7) =12a(8) =22a(9) =3a(10) =1a(11) =18a(12) =69a(13) =40a(14) =3a(15) =1a(16) =25a(17) =159a(18) =241a(19) =61a(20) =3a(21) =1a(22) =33a(23) =310a(24) =883a(25) =582a(26) =85a(27) =3a(28) =1a(29) =42