Array: row n shows the coefficients of the characteristic polynomial of the nth n-th principal submatrix of max(3i-j, 3j-i), as in A204156.

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 12 2012

proposed

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editing

proposed

allocated for Clark KimberlingArray: row n shows the coefficients of the characteristic polynomial of the nth principal submatrix of max(3i-j, 3j-i), as in A204156.

1, -1, -13, -4, 1, 88, 78, 9, -1, -496, -704, -260, -16, 1, 2560, 4960, 3080, 650, 25, -1, -12544, -30720, -26784, -9856, -1365, -36, 1, 59392, 175616, 197120, 104160, 25872, 2548, 49, -1, -274432, -950272, -1304576, -901120, -327360, -59136, -4368, -64, 1, 1245184, 4939776, 8017920, 6849024, 3294720

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

-13...-4.....1

88....78....9.....-1

-496..-704..-260...-16...1

f[i_, j_] := -1 + Max[3 i - j, 3 j - 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}]] (* A204156 *)

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[%] (* A204157 *)

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

Cf. A204156, A202605, A204016.

allocated

tabl,sign

Clark Kimberling (ck6(AT)evansville.edu), Jan 12 2012

approved

editing

allocated for Clark Kimberling

allocated

approved