|
%I #11 Feb 13 2023 03:05:08
%S 1,-1,0,-2,1,-1,-1,3,-1,-2,2,3,-4,1,-3,7,-2,-6,5,-1,-4,14,-15,0,10,-6,
%T 1,-5,23,-39,25,5,-15,7,-1,-6,34,-77,84,-35,-14,21,-8,1,-7,47,-132,
%U 196,-154,42,28,-28,9,-1,-8,62,-207,384,-420,252
%N 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 i=1 or j=1 or i=j, and 0 otherwise) as in A204177.
%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.)
%e Top of the array:
%e 1...-1
%e 0...-2.....1
%e -1...-1.....3...-1
%e -2....2.....3...-4...1
%e -3....7....-2...-6...5...-1
%t f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 1;
%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}]] (* A204177 *)
%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[%] (* A204178 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204177, A202605, A204016.
%K tabf,sign
%O 1,4
%A _Clark Kimberling_, Jan 12 2012
|
