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

Terms

a(0) =1a(1) =2a(2) =1a(3) =5a(4) =4a(5) =1a(6) =16a(7) =17a(8) =7a(9) =1a(10) =65a(11) =84a(12) =45a(13) =11a(14) =1a(15) =326a(16) =485a(17) =309a(18) =100a(19) =16a(20) =1a(21) =1957a(22) =3236a(23) =2339a(24) =909a(25) =196a(26) =22a(27) =1a(28) =13700a(29) =24609