A168676 Coefficients of characteristic polynomials for a two diagonal Matrix type that has determinant equal to trace:M(n)=Table[If[ k == m && m < n, 1, If[k == m + 1, 1, If[k == 1 && m == n, (-1)^(n + 1)*(n - 1), 0]]], {k, n}, {m, n}]

-1, 0, -1, 1, -1, 1, 2, -1, 2, -1, 3, -1, 3, -3, 1, 4, -1, 4, -6, 4, -1, 5, -1, 5, -10, 10, -5, 1, 6, -1, 6, -15, 20, -15, 6, -1, 7, -1, 7, -21, 35, -35, 21, -7, 1, 8, -1, 8, -28, 56, -70, 56, -28, 8, -1, 9, -1, 9, -36, 84, -126, 126, -84, 36, -9, 1

Offset

0,7

Comments

This set of matrices was constructed so that the determinant would be equal to the trace.

This system is interesting because the result contains a signed Pascal's triangle.

The initial term is adjusted to {-1} to fit the first column.

Row sums are:{-1, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9,...}.

Links

Table of n, a(n) for n=0..65.

Example

{-1},
{0, -1},
{1, -1, 1},
{2, -1, 2, -1},
{3, -1, 3, -3, 1},
{4, -1, 4, -6, 4, -1},
{5, -1, 5, -10, 10, -5, 1},
{6, -1, 6, -15, 20, -15, 6, -1},
{7, -1, 7, -21, 35, -35, 21, -7, 1},
{8, -1, 8, -28, 56, -70, 56, -28, 8, -1},
{9, -1, 9, -36, 84, -126, 126, -84, 36, -9, 1}

Mathematica

Clear[M, n, m, k]
M[n_] := Table[If[ k == m && m < n, 1, If[k == m + 1, 1, If[k == 1 && m == n, (-1)^(n + 1)*(n - 1), 0]]], {k, n}, {m, n}]
TableForm[Table[M[n], {n, 1, 10}]]
Table[Det[M[n]], {n, 1, 10}]
Table[Sum[M[n][[k, k]], {k, 1, n}], {n, 1, 10}]
a = Join[{{-1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 10}]]
Flatten[a]

Crossrefs

Sequence in context: A027351 A029349 A372205 * A202327 A243546 A243554

Adjacent sequences: A168673 A168674 A168675 * A168677 A168678 A168679

Keyword

uned,sign

Author

Roger L. Bagula, Dec 02 2009

Status

approved