A204017 - OEIS

%I #6 Jul 12 2012 00:39:54

%S 0,-1,-1,0,1,4,6,0,-1,-15,-38,-20,0,1,56,206,184,50,0,-1,-185,-1072,

%T -1357,-630,-105,0,1,204,5146,9276,6060,1736,196,0,-1,6209,-17334,

%U -58470,-52452,-21102,-4116,-336,0,1,-112400,-67682,293984

%N 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 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 (For references regarding interlacing roots, see A202605.)

%e Top of the array:

%e  1... -1

%e -1.... 0.... 1

%e  4.... 6.... 0... -1

%e -15.. -38.. -20... 0... 1

%e  56... 206.. 184.. 50.. 0.. -1

%e ...

%e The 1st principal submatrix (ps) of A204016 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

%e ...

%e The 2nd ps is {{0,1},{1,0}}, with p(2)=-1+x^2 and zero-set {-1,1}.

%e ...

%e 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 ...

%e 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 ...

%e The interlace property is illustrated for the last two zero-sets by this chain:

%e -3 < -2 < -1.7 < -0.7 < -0.5 < 2.7 < 5.2

%t f[i_, j_] := Max[Mod[i, j], Mod[j, i]];

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t {n, 1, 12}, {i, 1, n}]]  (* A204016 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]               (* A204017 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204016, A202605.

%K tabl,sign

%O 1,6

%A _Clark Kimberling_, Jan 10 2012