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A204019
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{1+j mod i, 1+i mod j} (A204018).
3
1, -1, -3, -2, 1, 8, 14, 3, -1, -21, -64, -40, -4, 1, 40, 266, 280, 90, 5, -1, 125, -930, -1671, -896, -175, -6, 1, -2940, 854, 8600, 7228, 2352, 308, 7, -1, 35035, 37744, -27334, -50164, -24594, -5376, -504, -8, 1, -372400
OFFSET
1,3
COMMENTS
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). The least zero of p(n) is -n.
For n>1, the least zero of p(n) is exactly 1-n; the greatest, for p(1) to p(5) is represented by (1,3,5.701...,9.158...13.392...).
See A202605 and A204016 for guides to related sequences.
REFERENCES
(For references regarding interlacing roots, see A202605.)
EXAMPLE
Top of the array:
1....-1
-3....-2......1
8.....14.....3....-1
-21...-64....-40...-4...1
MATHEMATICA
f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204018 *)
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[%] (* A204019 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved