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A202019 Triangle by rows, related to the numbers of binary trees of height less than n, derived from the Mandelbrot set. 3
1, 1, 1, 1, 2, 1, 1, 1, 4, 6, 6, 5, 2, 1, 1, 1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1, 1, 16, 120, 568, 1932, 5096, 10948, 19788, 30782, 41944, 50788, 55308, 54746, 49700, 41658, 32398, 23461, 15864, 10068, 6036, 3434, 1860, 958, 470, 221, 100, 42, 14, 5, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

As shown on p. 74 [Diaconis & Graham], n-th row polynomials are cyclic with period n, given real roots, if the polynomials are divided through by n. For example, taking x^3 + 2x^2 + x + 1 = 0, the real root = -1.75487766... = c. Then using x^2 + c, we obtain the period three trajectory: -1.75487... -> 1.32471...-> 0.

The shuffling connection [p.75], resulting in a permutation that is the Gilbreath shuffle: "To make the connection with shuffling cards, write down a periodic sequence starting at zero. Write a one above the smallest point, a two above the next smallest point and so on. For example, if c = -1.75486...(a period three point), we have:

2.............1.............3......

0........-1.75487........1.32471... For a fixed value of c, the numbers written on top code up a permutation that is a Gilbreath shuffle".

Row sums = A003095: (1, 2, 5, 26, 677,...) relating to the number of binary trees of height less than n.

Let f(z) = z^2 + c, then row k lists the expansion of the n-fold composition f(f(...f(0)...) in falling powers of c (with the coefficients for c^0 omitted). The n initial terms of the reversed n-th row are the Catalan numbers (cf. A137560). - Joerg Arndt, Jun 04 2016

REFERENCES

Persi Diaconis & R. L. Graham, "Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks", Princeton University Press, 2012; pp. 73-83.

LINKS

Alois P. Heinz, Rows n = 1..13, flattened

FORMULA

Coefficients of x by rows such that given n-th row p(x), the next row is (p(x))^2 + x); starting x -> (x^2 + x) -> (x^4 + 2x^3 + x^2 + x)...

T(n,k) = A309049(2^(n-1)-1,k-1). - Alois P. Heinz, Jul 11 2019

EXAMPLE

Row 4 = (1, 4, 6, 6, 5, 2, 1, 1) since (x^4 + 2x^3 + x^2 + x)^2 + x = x^8 + 4x^7 + 6x^6 + 6x^5 + 5x^4 + 2x^3 + x^2 + x.

Triangle begins:

  1;

  1, 1;

  1, 2,  1,  1;

  1, 4,  6,  6,  5,   2,   1,  1;

  1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1;

  ...

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-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2^(n-1)-1)):

seq(T(n), n=1..7);  # Alois P. Heinz, Jul 11 2019

MATHEMATICA

b[n_] := b[n] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f]*

     b[n-1-f]]][Min[g-1, n-g/2]]][2^(Length[IntegerDigits[n, 2]]-1)]];

T[n_] := CoefficientList[b[2^(n-1)-1], x];

Array[T, 7] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)

CROSSREFS

Row sums are A003095.

Cf. A137560 (reversed rows).

Cf. A309049.

Sequence in context: A095231 A303697 A342413 * A295685 A330942 A141471

Adjacent sequences:  A202016 A202017 A202018 * A202020 A202021 A202022

KEYWORD

nonn,tabf

AUTHOR

Gary W. Adamson, Dec 08 2011

STATUS

approved

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Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)