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) =6a(6) =1a(7) =9a(8) =21a(9) =12a(10) =1a(11) =12a(12) =45a(13) =60a(14) =24a(15) =1a(16) =15a(17) =78a(18) =171a(19) =156a(20) =48a(21) =1a(22) =18a(23) =120a(24) =372a(25) =558a(26) =384a(27) =96a(28) =1a(29) =21