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 A204167 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164. 3

%I

%S 1,-1,-2,-3,1,1,6,6,-1,0,-4,-16,-10,1,0,0,15,32,15,-1,0,0,0,-36,-60,

%T -21,1,0,0,0,0,84,100,28,-1,0,0,0,0,0,-160,-160,-36,1,0,0,0,0,0,0,300,

%U 240,45,-1,0,0,0,0,0,0,0,-500,-350

%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

%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 -2....-3.....1

%e 1.....6.....6....-1

%e 0....-4....-16...-10...1

%t f[i_, j_] := Ceiling[(i + j)/2];

%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}]] (* A204166 *)

%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[%] (* A204167 *)

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

%Y Cf. A204166, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 12 2012

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Last modified March 28 14:02 EDT 2020. Contains 333089 sequences. (Running on oeis4.)