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Search: id:A123948
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| A123948 |
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Bernstein-Bezier type polynomials are made in to matrices closely related to Pasdcal's triangle and binomial expansions: the characteristic polynomials of the resulting matrices give the triangular sequence. |
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+0
1
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| 1, 1, -1, -1, 1, 1, -2, 3, 3, -1, 9, -15, -22, 7, 1, 96, -184, -314, 139, 19, -1, -2500, 5250, 10575, -5375, -1026, 51, 1, -162000, 369900, 842310, -498171, -111179, 7644, 141, -1, 26471025, -64790985, -164634169, 109325076, 28870212, -2322404, -59193, 393, 1, 11014635520, -28677390336
(list; table; graph; listen)
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OFFSET
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1,7
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COMMENT
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Matrices: ( lower triangular form) 1 X 1 {{1}}, 2 X 2 {{0, 1}, {1, -1}}, 3 X 3 {{0,0, 1}, {0, 2, -2}, {1, -2, 1}}, 4 X 4 {{0, 0, 0, 1}, {0, 0,3, -3}, {0, 3, -6, 3}, {1, -3, 3, -1}}, 5 X 5 {{0, 0, 0, 0, 1}, {0, 0, 0, 4, -4}, {0, 0, 6, -12, 6}, {0, 4, -12, 12, -4}, {1, -4, 6, -4, 1}}, 6 X 6 {{0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 5, -5}, {0, 0, 0, 10, -20, 10}, {0, 0, 10, -30, 30, -10}, {0, 5, -20, 30, -20, 5}, {1, -5, 10, -10, 5, -1}}
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REFERENCES
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Over and Over Again, Chang and Sederberg, MAA, 1997 cxhapter 30
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FORMULA
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p(n,i,x)=binomial[n, n - i]*(1 - x)^i*x^(n - i) a(i,j)=CoefficientList[p(n,i,x)] p'(n,x)=CharacteristicPolynomial(a(i,j)) p'(n,x)->t(n,m)
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EXAMPLE
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Triangle begins:
{1},
{1, -1},
{-1,1, 1},
{-2, 3, 3, -1},
{9, -15, -22, 7, 1},
{96, -184, -314, 139, 19, -1},
{-2500, 5250, 10575, -5375, -1026, 51, 1},
{-162000,369900, 842310, -498171, -111179, 7644, 141, -1}
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MATHEMATICA
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M[n_] := Table[CoefficientList[Binomial[n, n - i]*(1 - x)^i*x^(n - i), x], {i, 0, n}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 0, 10}]]; Flatten[a]
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CROSSREFS
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Sequence in context: A116155 A097005 A068008 this_sequence A131012 A083057 A099028
Adjacent sequences: A123945 A123946 A123947 this_sequence A123949 A123950 A123951
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KEYWORD
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uned,probation,tabl,sign
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AUTHOR
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Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 26 2006
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