

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.
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 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



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)):


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.


KEYWORD



AUTHOR



EXTENSIONS

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


STATUS

approved



