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