Triangular array: Row n shows the coefficients of polynomials p(n,x) defined in Comments.

Terms

a(0) =1a(1) =4a(2) =1a(3) =5a(4) =4a(5) =1a(6) =24a(7) =11a(8) =4a(9) =1a(10) =35a(11) =52a(12) =18a(13) =4a(14) =1a(15) =192a(16) =123a(17) =84a(18) =26a(19) =4a(20) =1a(21) =315a(22) =660a(23) =285a(24) =120a(25) =35a(26) =4a(27) =1a(28) =1920a(29) =1545