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Search: id:A123199
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| A123199 |
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A special Bezier transform of the binomial triangle sequence using: x-> 2*x 1-x-> 1-x^2 t(n,m,x)=binomial[m, n]*(2*x)^n*(1-x^2)^(m - n). |
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+0
1
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| 1, 1, 2, -1, 1, 4, 2, -4, 1, 1, 6, 9, -4, -9, 6, -1, 1, 8, 20, 8, -26, -8, 20, -8, 1, 1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1, 1, 12, 54, 100, 15, -168, -76, 168, 15, -100, 54, -12, 1, 1, 14, 77, 196, 161, -238, -427, 184, 427, -238, -161, 196, -77, 14, -1, 1, 16, 104, 336, 476, -112, -1064, -432, 1222, 432, -1064, 112, 476
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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The ordinary Bezier transform is degenerate to 1 in all cases. This transform is pased on (x,1-x}->{2*x/(1+x^2),(1-x^2)/(1+x^2)} with has a sum of squares equal to one. This sequence is based on the similarity of Poncelet's trajectories using {2*x/(1+x^2),(1-x^2)/(1+x^2)} to de Casteljau's construction for Bezier-Bernstein Polynomials using (x,1-x}.
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REFERENCES
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Elliptic Curves, McKean and Moll, 1997, Cambridge University Press, page 106, figure 2.22
Over and Over Again, Chang and Sederberg,MAA,1997, page164, figure 26.1
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FORMULA
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Coeffiencts of: t(n,m,x)=binomial[m, n]*(2*x)^n*(1-x^2)^(m - n)
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EXAMPLE
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1
1, 2, -1
1, 4, 2, -4, 1
1, 6, 9, -4, -9, 6, -1
1, 8, 20, 8,-26, -8, 20,-8, 1
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MATHEMATICA
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x1 = 2*x; x2 = (1 - x^2); a = Table[Expand[Sum[Binomial[m, n]*x1^n*x2^(m - n), {n, 0, m}]], {m, 0, 10}]; w = Table[CoefficientList[a[[n]], x], {n, 1, 10}] Flatten[w]
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CROSSREFS
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Sequence in context: A129033 A054090 A122517 this_sequence A157125 A157143 A112096
Adjacent sequences: A123196 A123197 A123198 this_sequence A123200 A123201 A123202
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KEYWORD
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uned,sign
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 04 2006
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