Array: row n shows the coefficients of the characteristic polynomial of the nth n-th principal submatrix of f(i,j)=(L(i) if i=j and 1 otherwise) (A204129).

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)=(L(i) if i=j and 1 otherwise) (A204129).

1, -1, 2, -4, 1, 6, -16, 8, -1, 36, -108, 69, -15, 1, 360, -1152, 834, -230, 26, -1, 6120, -20304, 15726, -4890, 693, -44, 1, 171360, -580752, 467724, -155524, 24797, -1963, 73, -1, 7882560, -27057312, 22300752, -7709504

1,3

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

2....-4.....1

6....-16....8....-1

36...-108...69...-15...1

f[i_, j_] := 1; f[i_, i_] := LucasL[i];

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}]] (* A204129 *)

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[%] (* A204130 *)

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

Cf. A204129, A202605, A204016.

allocated

tabl,sign

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

approved

editing

allocated for Clark Kimberling

allocated

approved