cp's OEIS Frontend

A204172 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is odd, and 0 otherwise) as in A204171.

%I A204172 #17 Feb 10 2023 18:51:13
%S A204172 1,-1,0,-1,1,-1,1,2,-1,0,1,-1,-2,1,1,-1,-4,3,3,-1,0,-1,1,4,-3,-3,1,-1,
%T A204172 1,6,-5,-10,6,4,-1,0,1,-1,-6,5,10,-6,-4,1,1,-1,-8,7,21,-15,-20,10,5,
%U A204172 -1,0,-1,1,8,-7,-21,15,20,-10,-5,1,-1,1,10,-9,-36,28,56
%N A204172 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is odd, and 0 otherwise) as in A204171.
%C A204172 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.
%C A204172 This sequence uses the characteristic polynomial defined as det(A - x I), rather than det(x I - A), so the last term in row n is (-1)^n. - _Robert Israel_, Feb 10 2023
%D A204172 (For references regarding interlacing roots, see A202605.)
%H A204172 Robert Israel, <a href="/A204172/b204172.txt">Table of n, a(n) for n = 1..10010</a> (rows 1 to 140, flattened)
%F A204172 (Empirical) T(m,k) = [x^m y^(k-1)] y*(1-x*y)*(1-x+x^3*y^2)/(1+y^2-2*x^2*y^2+x^4*y^4). - _Robert Israel_, Feb 10 2023
%e A204172 Top of the array:
%e A204172    1, -1;
%e A204172    0, -1,  1;
%e A204172   -1,  1,  2, -1;
%e A204172    0,  1, -1, -2, 1;
%p A204172 for n from 1 to 20 do
%p A204172   P:= (-1)^n * LinearAlgebra:-CharacteristicPolynomial(Matrix(n,n,(i,j) -> max(i,j) mod 2),x):
%p A204172   print(seq(coeff(P,x,i),i=0..n));
%p A204172 od: # _Robert Israel_, Feb 10 2023
%t A204172 f[i_, j_] := If[Mod[Max[i, j], 2] == 1, 1, 0]
%t A204172 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204172 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204172 Flatten[Table[f[i, n + 1 - i],
%t A204172   {n, 1, 15}, {i, 1, n}]]  (* A204171 *)
%t A204172 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204172 c[n_] := CoefficientList[p[n], x]
%t A204172 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204172 Table[c[n], {n, 1, 12}]
%t A204172 Flatten[%]                 (* A204172 *)
%t A204172 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204172 Cf. A204171, A202605, A204016.
%K A204172 tabf,sign
%O A204172 1,8
%A A204172 _Clark Kimberling_, Jan 12 2012