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%I #49 Oct 27 2021 01:00:08
%S 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,
%T 116,94,60,28,8,1,0,1,1,2,5,14,42,100,221,470,958,1860,3434,6036,
%U 10068,15864,23461,32398,41658,49700,54746,55308,50788,41944,30782,19788
%N 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.
%C The root of one of these polynomials gives Julia Douady's rabbit.
%C These polynomials are basic to the theory of "cycles" in complex dynamics.
%C These polynomials are also described in a comment by Donald D. Cross in the entry for the Catalan numbers, A000108.
%C Except for the first row, row sums are A003095 (a(n) = a(n-1)^2 + 1). - _Gerald McGarvey_, Sep 26 2008
%C 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
%C 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
%C The sequence of polynomials is conjectured to tend to the Catalan numbers (A000108). - _Jon Perry_, Oct 31 2010
%C It can be shown that the initial n nonzero terms of row n are the first Catalan numbers. - _Joerg Arndt_, Jun 04 2016
%C From _Jianing Song_, Mar 23 2021: (Start)
%C 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.
%C 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)
%D Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 128-129
%H Alois P. Heinz, <a href="/A137560/b137560.txt">Rows n = 0..13, flattened</a> (rows n=0..8 from Roger L. Bagula)
%H Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, <a href="https://ojs.lib.uwo.ca/index.php/maple/article/view/14037">Some Facts and Conjectures about Mandelbrot Polynomials</a>, Maple Trans., Vol. 1, No. 1, Article 1 (July 2021).
%H Robert Munafo, <a href="http://www.mrob.com/pub/muency/lemniscates.html">Lemniscates</a> [From _Robert Munafo_, Oct 29 2009]
%H Everything2 user ferrouslepidoptera, <a href="http://www.everything2.com/index.pl?node_id=710894&lastnode_id=0">How many melodies are there in the universe?</a> [From _Robert Munafo_, Oct 29 2009]
%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Mandelbrot_set">Mandelbrot set</a>
%e Triangle starts:
%e {1},
%e {0, 1},
%e {0, 1, 1},
%e {0, 1, 1, 2, 1},
%e {0, 1, 1, 2, 5, 6, 6, 4, 1},
%e {0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1},
%e {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},
%e ...
%p b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
%p x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
%p end:
%p T:= n-> `if`(n=0, 1, (m-> (p-> seq(coeff(p, x, m-i),
%p i=-1..m))(b(m)))(2^(n-1)-1)):
%p seq(T(n), n=0..7); # _Alois P. Heinz_, Jul 11 2019
%t 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]}];
%o (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
%Y A052154 gives the same array read by antidiagonals.
%Y A137867 gives the related Misiurewicz polynomials. [From _Robert Munafo_, Dec 12 2009]
%Y Cf. A202019 (reversed rows).
%Y Cf. A309049.
%K nonn,tabf,look
%O 0,10
%A _Roger L. Bagula_, Apr 25 2008
%E Edited by _N. J. A. Sloane_, Apr 26 2008
%E Offset set to 0 and new name from _Joerg Arndt_, Jun 04 2016
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