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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
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, p. 133ff.
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: A335391 A014674 A015339 * A324734 A359902 A111143
KEYWORD
tabl,uned,sign
AUTHOR
Roger L. Bagula, Apr 29 2008
STATUS
approved