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, 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.

Links

Table of n, a(n) for n=1..109.

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

Adjacent sequences: A137864 A137865 A137866 * A137868 A137869 A137870

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 29 2008

Status

approved