A122188 Triangle read by rows, formed from the coefficients of characteristic polynomials of the following sequence of matrices: 2 X 2 {{0, 1}, {1, 1}}, 3 X 3 {{0, 1, 0}, {0, 0, 1}, {1, 1, 1}}, 4 X 4 {{0, 1,0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 1, 1}}, 5 X 5 {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 1, 1, 1, 1}}, ...

1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,1

Comments

Row sums are {1, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9}.

Links

Table of n, a(n) for n=1..104.

Formula

B(x, n) = If[n > 1, (-1)^n*(x^n - Sum[x^m, {m, 0, n - 1}])]

Example

Triangular array:
{1},
{1, -1},
{-1, -1, 1},
{1, 1, 1, -1},
{-1, -1, -1, -1, 1},
{1, 1, 1,1, 1, -1},
{-1, -1, -1, -1, -1, -1,1},
{1, 1, 1, 1, 1, 1, 1, -1},
{-1, -1, -1, -1, -1, -1, -1, -1, 1}

Mathematica

An[d_] := Table[If[n == d, 1, If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
Clear[B, x, n] B[x, 0] = 1; B[x, 1] = -x + 1; B[x_, n_] := B[x, n] = If[n > 1, (-1)^n*(x^n - Sum[x^m, {m, 0, n - 1}])]; Table[ExpandAll[B[x, n]], {n, 0, 10}]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[B[x, n], x]], {n, 0, 10}]

Crossrefs

Cf. A000073, A000045.

Sequence in context: A065357 A119665 A121241 * A158388 A162285 A186035

Adjacent sequences: A122185 A122186 A122187 * A122189 A122190 A122191

Keyword

tabl,sign

Author

Gary W. Adamson and Roger L. Bagula, Oct 18 2006, Mar 18 2008

Extensions

Edited by N. J. A. Sloane, Apr 14 2008

Status

approved