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Search: id:A137560
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| A137560 |
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Triangle read by rows: coefficients of the Mandelbrot-Julia quadratic polynomials (sometimes called cycle polynomials or Pc polynomials): Pc[0]=1; Pc[1]=c; Pc[n]=nestedfunction(Pc[z^2+c],n]. |
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2
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| 1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 6, 6, 4, 1, 0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1, 0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788
(list; table; graph; listen)
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OFFSET
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1,10
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COMMENT
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Row sums are: {1, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, ...};
The root of one of these polynomials g1ves Julia Douady's rabbit.
These polynomials are basic to the theory of "cycles" in complex dynamics.
These polynomials are also described in a comment by Donald D. Cross in the entry for the Catalan numbers, A000108.
Except for the first row, row sums are A003095 (a(n) = a(n-1)^2 + 1). [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Sep 26 2008]
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REFERENCES
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Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer,New York,1993,pp 128-129
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LINKS
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Roger L. Bagula, Table of n, a(n) for n = 1..264
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EXAMPLE
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Examples: Pc[2] = f[z,c] = z^2+c with z->c = c^2+c; Pc[3] = f[f[z,c],c] => c+c^2+2*c^3+c^4.
{1},
{0, 1},
{0, 1, 1},
{0, 1, 1, 2, 1},
{0, 1, 1, 2, 5, 6, 6, 4, 1},
{0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
{0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1},
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MATHEMATICA
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f[z_] = z^2 + x; g = Join[{1}, ExpandAll[NestList[f, x, 7]]]; a = Table[CoefficientList[g[[n]], x], {n, 1, Length[g]}]; Flatten[a] Table[Apply[Plus, CoefficientList[g[[n]], x]], {n, 1, Length[g]}];
The following Mma program gives a second method for computing the polynomials without using a nesting method: p(x, n)=p(x, n-1)^2+x.
p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = x; p[x, 2] = x + x^2; p[x_, n_] := p[x, n] = p[x, n - 1]^2 + x; Table[ExpandAll[p[x, n]], {n, 0, 7}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 7}]; Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 7}];
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PROGRAM
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(PARI) p = vector(6); p[1] = x; for(n=2, 6, p[n] = p[n-1]^2 + x); print1("1"); for(n=1, 6, for(m=0, poldegree(p[n]), print1(", ", polcoeff(p[n], m)))) [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Sep 26 2008]
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CROSSREFS
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Sequence in context: A035440 A029878 A080934 this_sequence A131255 A133607 A103631
Adjacent sequences: A137557 A137558 A137559 this_sequence A137561 A137562 A137563
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KEYWORD
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nonn,tabl
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 25 2008, Apr 27 2008
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EXTENSIONS
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Edited by njas, Apr 26 2008
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