login
A204170
Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of (i*j), as in A003991.
2
1, -1, 0, -5, 1, 0, 0, 14, -1, 0, 0, 0, -30, 1, 0, 0, 0, 0, 55, -1, 0, 0, 0, 0, 0, -91, 1, 0, 0, 0, 0, 0, 0, 140, -1, 0, 0, 0, 0, 0, 0, 0, -204, 1, 0, 0, 0, 0, 0, 0, 0, 0, 285, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -385, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 506, -1
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.
p(n,x) = x^n + (-1)^n*s(n)*x^n - 1, where s=A000330 (square pyramidal numbers).
REFERENCES
(For references regarding interlacing roots, see A202605.)
EXAMPLE
Top of the array:
1, -1;
0, -5, 1;
0, 0, 14, -1;
0, 0, 0, -30, 1;
MATHEMATICA
f[i_, j_] := i*j;
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}]] (* A003991 *)
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[%] (* A204170 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Clark Kimberling, Jan 12 2012
STATUS
approved