|
| |
|
|
A158286
|
|
A triangle of coefficients of matrix polynomials:M(d)=Table[If[ m == n, d, -1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, -d, 1], {n, 1, d}, {m, 1, d}]; (strange designs)
|
|
0
| |
|
|
1, -1, -1, 9, 10, 1, -256, -288, -33, -1, 15625, 17500, 1950, 76, 1, -1679616, -1866240, -194400, -7920, -145, -1, 282475249, 311299254, 30000495, 1200500, 24255, 246, 1, -68719476736, -75161927680, -6694109184, -256901120, -5304320
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Row sums are:
{1, -2, 3, 0, -27, 162, -729, 2916, -10935, 39366, -137781,...}.
These matrices are products an all ones matrix with central diagonal negative n.
These matrices have the strange design orthogonality property of:
M(n)-(2-n)*Ones(n)=(n+1)^2*IdentityMatirix[n];
Table[If[ m == n, d, -1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, -d, 1], {n, 1, d}, {m, 1, d}]
- (2 + d)*Table[1, {n, d}, {m, d}]
Matrix example:
M(3)={{-11, 5, 5},
{5, -11, 5},
{5, 5, -11}}.
|
|
|
FORMULA
| M(d)=Table[If[ m == n, d, -1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, -d, 1], {n, 1, d}, {m, 1, d}];
out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)
|
|
|
EXAMPLE
| {1},
{-1, -1},
{9, 10, 1},
{-256, -288, -33, -1},
{15625, 17500, 1950, 76, 1},
{-1679616, -1866240, -194400, -7920, -145, -1},
{282475249, 311299254, 30000495, 1200500, 24255, 246, 1},
{-68719476736, -75161927680, -6694109184, -256901120, -5304320, -61824, -385, -1},
{22876792454961, 24853799210328, 2050229227788, 74729107656, 1525235670, 18738216, 138348, 568, 1},
{-10000000000000000, -10800000000000000, -828000000000000, -28560000000000, -567000000000, -7056000000, -56280000, -280800, -801, -1},
{5559917313492231481, 5973464882264380930, 427218562781146125, 13934621992641240, 266896099144290, 3295124718732, 27158030130, 149338200, 528165, 1090, 1}
|
|
|
MATHEMATICA
| Clear[M, T, d, a, x, a0];
M[d_] := Table[If[ m == n, d, -1], {n, 1, d}, {m, 1, d}].Table[If[ m == n, -d, 1], {n, 1, d}, {m, 1, d}];
a0 = Table[M[d], {d, 1, 10}];
Table[Det[M[d]], {d, 1, 10}];
Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
a = Join[{{1}}, Table[CoefficientList[Expand[ CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
Flatten[a]
Join[{1}, Table[ Apply[Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];
|
|
|
CROSSREFS
| Sequence in context: A067450 A109409 A160563 * A126839 A034058 A205151
Adjacent sequences: A158283 A158284 A158285 * A158287 A158288 A158289
|
|
|
KEYWORD
| sign,tabl,uned
|
|
|
AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2009
|
| |
|
|
