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

Terms

a(0) =1a(1) =3a(2) =1a(3) =12a(4) =5a(5) =60a(6) =28a(7) =1a(8) =360a(9) =180a(10) =11a(11) =2520a(12) =1320a(13) =105a(14) =1a(15) =20160a(16) =10920a(17) =1020a(18) =19a(19) =181440a(20) =100800a(21) =10500a(22) =276a(23) =1a(24) =1814400a(25) =1028160a(26) =115920a(27) =3780a(28) =29a(29) =19958400