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A204155 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.

%I A204155 #13 Dec 04 2017 02:55:16
%S A204155 1,-1,-7,-3,1,33,39,6,-1,-135,-255,-125,-10,1,513,1323,1092,305,15,-1,
%T A204155 -1863,-6075,-7047,-3444,-630,-21,1,6561,25839,38610,27135,8946,1162,
%U A204155 28,-1,-22599,-104247,-190593,-175230
%N A204155 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.
%C A204155 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 A204155 (For references regarding interlacing roots, see A202605.)
%H A204155 Robert Israel, <a href="/A204155/b204155.txt">Table of n, a(n) for n = 1..10010</a> (rows 1 to 140, flattened)
%e A204155 Top of the array:
%e A204155      1,   -1;
%e A204155     -7,   -3,    1;
%e A204155     33,   39,    6,   -1;
%e A204155   -135, -255, -125,  -10,    1;
%p A204155 f:= proc(n) local P,lambda,i;
%p A204155 P:= (-1)^n*LinearAlgebra:-CharacteristicPolynomial(Matrix(n,n,(i,j) -> max(2*i-j,2*j-i)),lambda);
%p A204155 seq(coeff(P,lambda,i),i=0..n);
%p A204155 end proc:
%p A204155 map(f, [$1..20]); # _Robert Israel_, Dec 03 2017
%t A204155 f[i_, j_] := Max[2 i - j, 2 j - i];
%t A204155 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204155 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204155 Flatten[Table[f[i, n + 1 - i],
%t A204155   {n, 1, 15}, {i, 1, n}]]  (* A204154 *)
%t A204155 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204155 c[n_] := CoefficientList[p[n], x]
%t A204155 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204155 Table[c[n], {n, 1, 12}]
%t A204155 Flatten[%]                 (* A204155 *)
%t A204155 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204155 Cf. A204154, A202605, A204016.
%K A204155 tabl,sign,look
%O A204155 1,3
%A A204155 _Clark Kimberling_, Jan 12 2012