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) =2a(4) =3a(5) =6a(6) =10a(7) =1a(8) =24a(9) =42a(10) =7a(11) =120a(12) =216a(13) =45a(14) =1a(15) =720a(16) =1320a(17) =312a(18) =13a(19) =5040a(20) =9360a(21) =2400a(22) =136a(23) =1a(24) =40320a(25) =75600a(26) =20520a(27) =1400a(28) =21a(29) =362880