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A157785
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Coefficients of characteristic polynomials of a sequence of matrices:q=-2; m(n,k)=If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[ k == n && m > 1, 1, 0]]]].
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1
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1, 1, -1, -2, 1, 1, -8, 6, 3, -1, 64, -40, -30, 5, 1, 1024, -704, -440, 110, 11, -1, -32768, 21504, 14784, -3080, -462, 21, 1, -2097152, 1409024, 924672, -211904, -26488, 1806, 43, -1, 268435456, -178257920, -119767040, 26199040, 3602368, -204680
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Row sums except n=0 are zero.
Example:p(x,3)=-8 + 6 x + 3 x^2 - x^3);
M[3]={{4, 0, 0}, {-2, -2, 0}, {-2, 1, 1}}.
Known q's are:q=2 gives A135950;q=1 is a signed Pascal's triangle.
The matrix inverses seem to be related to the Gaussian q-form combinations.
Triangle T(n,k), 0<=k<=n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...](for q=-2)=[1,-3,4,-6,16,-36,64,...] DELTA [ -1,0,2,0,-4,0,8,0,-16,0,...] where DELTA is the operator defined in A084938 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 10 2009]
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FORMULA
| q=-2;
m(n,k)=If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[ k == n && m > 1, 1, 0]]]].
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EXAMPLE
| {1},
{1, -1},
{-2, 1, 1},
{-8, 6, 3, -1},
{64, -40, -30, 5, 1},
{1024, -704, -440, 110, 11, -1},
{-32768, 21504, 14784, -3080, -462, 21, 1},
{-2097152, 1409024, 924672, -211904, -26488, 1806, 43, -1},
{268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1},
{68719476736, -45902462976, -30482104320, 6826721280, 896007168, -56000448, -1666680, 29070, 171, -1},
{-35184372088832, 23433341566976, 15652739874816, -3464799191040, -465582391296, 27776222208, 909340608, -13217160, -116622, 341, 1}
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MATHEMATICA
| Clear[f, q, M, n, m];
q = -2;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]
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CROSSREFS
| A135950, A022166
Sequence in context: A077058 A053373 A102875 * A021476 A051428 A176698
Adjacent sequences: A157782 A157783 A157784 * A157786 A157787 A157788
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 06 2009
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