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A204017 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).

%I A204017 #6 Jul 12 2012 00:39:54
%S A204017 0,-1,-1,0,1,4,6,0,-1,-15,-38,-20,0,1,56,206,184,50,0,-1,-185,-1072,
%T A204017 -1357,-630,-105,0,1,204,5146,9276,6060,1736,196,0,-1,6209,-17334,
%U A204017 -58470,-52452,-21102,-4116,-336,0,1,-112400,-67682,293984
%N A204017 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i,  i mod j} (A204016).
%C A204017 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 A204016 and A202605 for guides to related sequences.
%D A204017 (For references regarding interlacing roots, see A202605.)
%e A204017 Top of the array:
%e A204017  1... -1
%e A204017 -1.... 0.... 1
%e A204017  4.... 6.... 0... -1
%e A204017 -15.. -38.. -20... 0... 1
%e A204017  56... 206.. 184.. 50.. 0.. -1
%e A204017 ...
%e A204017 The 1st principal submatrix (ps) of A204016 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A204017 ...
%e A204017 The 2nd ps is {{0,1},{1,0}}, with p(2)=-1+x^2 and zero-set {-1,1}.
%e A204017 ...
%e A204017 The 3rd ps is {{0,1,1},{1,0,2},{1,2,0}}, with p(3)=4+6x-x^3 and zero-set {-2, -0.732...,2.732...}.
%e A204017 ...
%e A204017 The 4th ps is {{0,1,1,1},{1,0,2,2},{1,2,0,3},{1,2,0,3}}, with p(4)=-15-38x-20x^2+x^4 and zero-set {-3, -1.714, -0.553, 5.268}.
%e A204017 ...
%e A204017 The interlace property is illustrated for the last two zero-sets by this chain:
%e A204017 -3 < -2 < -1.7 < -0.7 < -0.5 < 2.7 < 5.2
%t A204017 f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
%t A204017 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204017 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204017 Flatten[Table[f[i, n + 1 - i],
%t A204017 {n, 1, 12}, {i, 1, n}]]  (* A204016 *)
%t A204017 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204017 c[n_] := CoefficientList[p[n], x]
%t A204017 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204017 Table[c[n], {n, 1, 12}]
%t A204017 Flatten[%]               (* A204017 *)
%t A204017 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204017 Cf. A204016, A202605.
%K A204017 tabl,sign
%O A204017 1,6
%A A204017 _Clark Kimberling_, Jan 10 2012