Array: row n shows the coefficients of the characteristic polynomial of the nth n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

Let p(n)=p(n,x) be the characteristic polynomial of the nth 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.

_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 11 2012

proposed

approved

editing

proposed

allocated for Clark KimberlingArray: row n shows the coefficients of the characteristic polynomial of the nth principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 56, -268, 295, -119, 19, -1, 672, -3328, 3914, -1786, 361, -32, 1, 13440, -67904, 82936, -40496, 9237, -1027, 53, -1, 443520, -2267712, 2832024, -1437872, 350799, -43879, 2822

1,4

Let p(n)=p(n,x) be the characteristic polynomial of the nth 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.

(For references regarding interlacing roots, see A202605.)

Top of the array:

1....-1

1....-3.....1

2....-8.....6....-1

8....-36....35...-11...1

f[i_, j_] := 1; f[i_, i_] := Fibonacci[i + 1];

m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

TableForm[m[8]] (* 8x8 principal submatrix *)

Flatten[Table[f[i, n + 1 - i],

{n, 1, 15}, {i, 1, n}]] (* A204127 *)

p[n_] := CharacteristicPolynomial[m[n], x];

c[n_] := CoefficientList[p[n], x]

TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

Table[c[n], {n, 1, 12}]

Flatten[%] (* A204128 *)

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

Cf. A204127, A202605, A204016.

allocated

tabl,sign

Clark Kimberling (ck6(AT)evansville.edu), Jan 11 2012

approved

editing

allocated for Clark Kimberling

allocated

approved