Triangular array of coefficients of polynomials p(n,k) defined in Comments.

Terms

a(0) =1a(1) =2a(2) =2a(3) =5a(4) =12a(5) =9a(6) =16a(7) =68a(8) =112a(9) =64a(10) =65a(11) =420a(12) =1125a(13) =1375a(14) =625a(15) =326a(16) =2910a(17) =11124a(18) =21600a(19) =20736a(20) =7776a(21) =1957a(22) =22652a(23) =114611a(24) =311787a(25) =470596a(26) =369754a(27) =117649a(28) =13700a(29) =196872