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A204178
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if i=1 or j=1 or i=j, and 0 otherwise) as in A204177.
2
1, -1, 0, -2, 1, -1, -1, 3, -1, -2, 2, 3, -4, 1, -3, 7, -2, -6, 5, -1, -4, 14, -15, 0, 10, -6, 1, -5, 23, -39, 25, 5, -15, 7, -1, -6, 34, -77, 84, -35, -14, 21, -8, 1, -7, 47, -132, 196, -154, 42, 28, -28, 9, -1, -8, 62, -207, 384, -420, 252
OFFSET
1,4
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). See A202605 and A204016 for guides to related sequences.
REFERENCES
(For references regarding interlacing roots, see A202605.)
EXAMPLE
Top of the array:
1...-1
0...-2.....1
-1...-1.....3...-1
-2....2.....3...-4...1
-3....7....-2...-6...5...-1
MATHEMATICA
f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, 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}]] (* A204177 *)
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[%] (* A204178 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Clark Kimberling, Jan 12 2012
STATUS
approved