

A137560


Let f(z) = z^2 + c, then row k lists the expansion of the nfold composition f(f(...f(0)...) in rising powers of c.


7



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

0,10


COMMENTS

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.
Except for the first row, row sums are A003095 (a(n) = a(n1)^2 + 1).  Gerald McGarvey, Sep 26 2008
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
From Jianing Song, Mar 23 2021: (Start)
Let P_0(z) = 0, P_{n+1}(z) = P_n(z)^2 + z for n >= 0. For n > 0, the nth row gives the coefficients of P_n(z) (a polynomial with degree 2^(n1) 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(14*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(14*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)


REFERENCES

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 128129


LINKS

Alois P. Heinz, Rows n = 0..13, flattened (rows n=0..8 from Roger L. Bagula)
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Trans., Vol. 1, No. 1, Article 1 (July 2021).
Robert Munafo, Lemniscates [From Robert Munafo, Oct 29 2009]
Everything2 user ferrouslepidoptera, How many melodies are there in the universe? [From Robert Munafo, Oct 29 2009]
Wikipedia, Mandelbrot set


EXAMPLE

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},
...


MAPLE

b:= proc(n) option remember; `if`(n=0, 1, (g> (f> expand(
x^n+b(f)*b(n1f)))(min(g1, ng/2)))(2^ilog2(n)))
end:
T:= n> `if`(n=0, 1, (m> (p> seq(coeff(p, x, mi),
i=1..m))(b(m)))(2^(n1)1)):
seq(T(n), n=0..7); # Alois P. Heinz, Jul 11 2019


MATHEMATICA

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]}];


PROG

(PARI) p = vector(6); p[1] = x; for(n=2, 6, p[n] = p[n1]^2 + x); print1("1"); for(n=1, 6, for(m=0, poldegree(p[n]), print1(", ", polcoeff(p[n], m)))) \\ Gerald McGarvey, Sep 26 2008


CROSSREFS

A052154 gives the same array read by antidiagonals.
A137867 gives the related Misiurewicz polynomials. [From Robert Munafo, Dec 12 2009]
Cf. A202019 (reversed rows).
Cf. A309049.
Sequence in context: A288942 A294220 A214015 * A201093 A131255 A344566
Adjacent sequences: A137557 A137558 A137559 * A137561 A137562 A137563


KEYWORD

nonn,tabf,look


AUTHOR

Roger L. Bagula, Apr 25 2008


EXTENSIONS

Edited by N. J. A. Sloane, Apr 26 2008
Offset set to 0 and new name from Joerg Arndt, Jun 04 2016


STATUS

approved



