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A204144 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=max(ceiling(i/j),ceiling(j/i)) (as in A204143).

%I A204144 #6 Jul 12 2012 00:39:58
%S A204144 1,-1,-3,-2,1,8,14,3,-1,-12,-42,-35,-4,1,19,95,145,73,5,-1,-20,-140,
%T A204144 -338,-336,-125,-6,1,16,184,665,1037,735,205,7,-1,-16,-212,-981,-2140,
%U A204144 -2381,-1320,-303,-8,1,12,200,1209,3581,5727,5021
%N A204144 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=max(ceiling(i/j),ceiling(j/i)) (as in A204143).
%C A204144 Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 and A204016 for guides to related sequences.
%D A204144 (For references regarding interlacing roots, see A202605.)
%e A204144 Top of the array:
%e A204144  1...-1
%e A204144 -3...-2....1
%e A204144  8....14...3....-1
%e A204144 -12..-42..-35...-4....1
%t A204144 f[i_, j_] := Max[Ceiling[i/j], Ceiling[j/i]];
%t A204144 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204144 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204144 Flatten[Table[f[i, n + 1 - i],
%t A204144   {n, 1, 15}, {i, 1, n}]]  (* A204143 *)
%t A204144 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204144 c[n_] := CoefficientList[p[n], x]
%t A204144 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204144 Table[c[n], {n, 1, 12}]
%t A204144 Flatten[%]                 (* A204144 *)
%t A204144 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204144 Cf. A204143, A202605, A204016.
%K A204144 tabl,sign
%O A204144 1,3
%A A204144 _Clark Kimberling_, Jan 11 2012