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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.

%I A137867 #9 Nov 24 2017 06:20:04
%S A137867 -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,
%T A137867 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,
%U A137867 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
%N 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.
%C A137867 Row sums are: {0, 1, 1, 3, 4, 4, 21, 24, 25, 25, 651, 672, 675, 676, 676, 457653, 458304, 458325, 458328, 458329, 458329};
%C A137867 The roots of these polynomials are called Misiurewicz points and they are found in the antenna areas of the Mandelbrot set M.
%D A137867 Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, p. 133ff.
%F A137867 Pc(x,n)-> Nested ( z^2+x: when z->x): A137560; Pc(x,n)-Pc(x,m); n<>m;
%e A137867 {-1, 1},
%e A137867 {0, 0, 1},
%e A137867 {-1, 1, 1},
%e A137867 {0, 0, 0, 2, 1},
%e A137867 {0, 0, 1, 2, 1},
%e A137867 {-1, 1, 1, 2, 1},
%e A137867 {0, 0, 0, 0, 4, 6, 6, 4, 1},
%e A137867 {0, 0, 0, 2, 5, 6, 6, 4, 1},
%e A137867 {0, 0, 1, 2, 5, 6, 6, 4, 1},
%e A137867 {-1, 1, 1, 2, 5, 6, 6, 4, 1},
%e A137867 {0, 0, 0, 0, 0, 8, 20, 40, 68, 94, 114, 116, 94, 60, 28, 8, 1},
%e A137867 {0, 0, 0, 0, 4, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
%e A137867 {0, 0, 0, 2,5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
%e A137867 {0, 0, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
%e A137867 {-1, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
%e A137867 {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},
%e A137867 {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},
%e A137867 {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},
%e A137867 {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},
%e A137867 {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},
%e A137867 {-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}
%t A137867 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]}];
%K A137867 tabl,uned,sign
%O A137867 1,12
%A A137867 _Roger L. Bagula_, Apr 29 2008