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A137867 Triangular sequence of coefficients of the Misiurewicz polynomial which are made from the Pc Mandelbrot -Julia polynomials A137560 as: Pc(x,n)-Pc(x,m); n<>m. 1
-1, 1, 0, 0, 1, -1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 2, 1, -1, 1, 1, 2, 1, 0, 0, 0, 0, 4, 6, 6, 4, 1, 0, 0, 0, 2, 5, 6, 6, 4, 1, 0, 0, 1, 2, 5, 6, 6, 4, 1, -1, 1, 1, 2, 5, 6, 6, 4, 1, 0, 0, 0, 0, 0, 8, 20, 40, 68, 94, 114, 116, 94, 60, 28, 8, 1, 0, 0, 0, 0, 4, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1, 0, 0, 0, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8 (list; table; graph; refs; listen; history; internal format)
OFFSET

1,12

COMMENTS

Row sums are:

{0, 1, 1, 3, 4, 4, 21, 24, 25, 25, 651, 672, 675, 676, 676, 457653, 458304, 458325, 458328, 458329, 458329};

The roots of these polynomials are called Misiurewicz points and they are found in the antenna areas of the Mandelbrot set M.

REFERENCES

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer,New York,1993, page 133 ff

FORMULA

Pc(x,n)-> Nested ( z^2+x: when z->x): A137560; Pc(x,n)-Pc(x,m); n<>m;

EXAMPLE

{-1, 1},

{0, 0, 1},

{-1, 1, 1},

{0, 0, 0, 2, 1},

{0, 0, 1, 2, 1},

{-1, 1, 1, 2, 1},

{0, 0, 0, 0, 4, 6, 6, 4, 1},

{0, 0, 0, 2, 5, 6, 6, 4, 1},

{0, 0, 1, 2, 5, 6, 6, 4, 1},

{-1, 1, 1, 2, 5, 6, 6, 4, 1},

{0, 0, 0, 0, 0, 8, 20, 40, 68, 94, 114, 116, 94, 60, 28, 8, 1},

{0, 0, 0, 0, 4, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},

{0, 0, 0, 2,5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},

{0, 0, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},

{-1, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},

{0, 0, 0, 0, 0, 0, 16, 56, 152, 376, 844, 1744, 3340, 5976, 10040, 15856, 23460, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1},

{0, 0, 0, 0, 0, 8, 36, 96, 220, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1},

{0, 0, 0, 0, 4, 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},

{0, 0, 0, 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},

{0, 0, 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},

{-1, 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}

MATHEMATICA

Clear[f, g, h, x]; f[z_] = z^2 + x; g = Join[{1}, ExpandAll[NestList[f, x, 5]]]; h = Union[Flatten[Table[Flatten[Table[If[n == m, {}, ExpandAll[g[[ n]] - g[[m]]]], {m, 1, n}]], {n, 1, Length[g]}]]]; a = Table[CoefficientList[h[[n]], x], {n, 1, Length[h]}]; Flatten[a] Table[Apply[Plus, CoefficientList[h[[n]], x]], {n, 1, Length[h]}];

CROSSREFS

Sequence in context: A029409 A014674 A015339 * A111143 A004197 A048571

Adjacent sequences:  A137864 A137865 A137866 * A137868 A137869 A137870

KEYWORD

tabl,uned,sign

AUTHOR

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 29 2008

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Last modified February 15 11:53 EST 2012. Contains 205778 sequences.