A137560 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.

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

Offset: 0

Author

Roger L. Bagula, Apr 25 2008

Keywords

• nonn
• tabf
• look

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(n-1)^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 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)

Examples

			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},
  ...
		

References

• Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer, New York, 1993, pp 128-129

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

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.

Programs

Maple

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)):
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]}];

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

Extensions

Edited by N. J. A. Sloane, Apr 26 2008

Offset set to 0 and new name from Joerg Arndt, Jun 04 2016