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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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(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

REFERENCES

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

LINKS

Roger L. Bagula, Table of n, a(n) for n = 1..264

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]

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},

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[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

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).

Sequence in context: A240608 A080934 A214015 * A201093 A131255 A198295

Adjacent sequences:  A137557 A137558 A137559 * A137561 A137562 A137563

KEYWORD

nonn,tabf,look

AUTHOR

Roger L. Bagula, Apr 25 2008, Apr 27 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

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Last modified January 21 21:01 EST 2017. Contains 281110 sequences.