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A137560
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Let f(z) = z^2 + c, then row k lists the expansion of the n-fold composition f(f(...f(0)...) in rising powers of c.
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7
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1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 6, 6, 4, 1, 0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1, 0, 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
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OFFSET
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0,10
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COMMENTS
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The root of one of these polynomials gives Julia Douady's rabbit.
These polynomials are basic to the theory of "cycles" in complex dynamics.
These polynomials are also described in a comment by Donald D. Cross in the entry for the Catalan numbers, A000108.
The coefficients also enumerate the ways to divide a line segment into at most j pieces, with 0 <= j <= 2^n, in which every piece is a power of two in size (for example, 1/4 is allowed but 3/8 is not), no piece is less than 1/2^n of the whole, and every piece is aligned on a power of 2 boundary (so 1/4+1/2+1/4=1 is not allowed). See the everything2 web link (which treats the segment as a musical measure). - Robert Munafo, Oct 29 2009
Also the number of binary trees with exactly J leaf nodes and a height no greater than N. See the Munafo web page and note the connection to A003095. - Robert Munafo, Nov 03 2009
The sequence of polynomials is conjectured to tend to the Catalan numbers (A000108). - Jon Perry, Oct 31 2010
It can be shown that the initial n nonzero terms of row n are the first Catalan numbers. - Joerg Arndt, Jun 04 2016
Let P_0(z) = 0, P_{n+1}(z) = P_n(z)^2 + z for n >= 0. For n > 0, the n-th row gives the coefficients of P_n(z) (a polynomial with degree 2^(n-1) for n > 0) in rising powers of z. Note that the famous Mandelbrot set is Intersect_{n>=0} {z: |P_n(z)| <= 2}. In particular, the Mandelbrot set is compact since it is closed and bounded.
Let P(z) = (1 - sqrt(1-4*z))/2. For every 0 < r < 1/4, P_n(z) converges uniformly to P(z) on the disk {z: |z| <= r}, because |P_n(z) - P(z)| <= (1/2)*(1 - sqrt(1-4*r))^(n+1) for every |z| <= r. Note that P(z)/z is the generating function for Catalan numbers, which explains the comment from Joerg Arndt above. Is the convergence uniform on the disk {z: |z| <= 1/4}? (End)
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REFERENCES
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Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 128-129
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LINKS
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EXAMPLE
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Triangle starts:
{1},
{0, 1},
{0, 1, 1},
{0, 1, 1, 2, 1},
{0, 1, 1, 2, 5, 6, 6, 4, 1},
{0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
{0, 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},
...
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
end:
T:= n-> `if`(n=0, 1, (m-> (p-> seq(coeff(p, x, m-i),
i=-1..m))(b(m)))(2^(n-1)-1)):
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MATHEMATICA
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f[z_] = z^2 + x; g = Join[{1}, ExpandAll[NestList[f, x, 7]]]; a = Table[CoefficientList[g[[n]], x], {n, 1, Length[g]}]; Flatten[a] Table[Apply[Plus, CoefficientList[g[[n]], x]], {n, 1, Length[g]}];
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PROG
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(PARI) p = vector(6); p[1] = x; for(n=2, 6, p[n] = p[n-1]^2 + x); print1("1"); for(n=1, 6, for(m=0, poldegree(p[n]), print1(", ", polcoeff(p[n], m)))) \\ Gerald McGarvey, Sep 26 2008
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CROSSREFS
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A052154 gives the same array read by antidiagonals.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Offset set to 0 and new name from Joerg Arndt, Jun 04 2016
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STATUS
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approved
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