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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; text; 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, p. 133ff. LINKS 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 * A324734 A111143 A004197 Adjacent sequences:  A137864 A137865 A137866 * A137868 A137869 A137870 KEYWORD tabl,uned,sign AUTHOR Roger L. Bagula, Apr 29 2008 STATUS approved

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Last modified May 21 14:50 EDT 2019. Contains 323443 sequences. (Running on oeis4.)