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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.

%I A202019 #52 Feb 17 2025 01:31:32
%S A202019 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,
%T A202019 2,1,1,1,16,120,568,1932,5096,10948,19788,30782,41944,50788,55308,
%U A202019 54746,49700,41658,32398,23461,15864,10068,6036,3434,1860,958,470,221,100,42,14,5,2,1,1
%N A202019 Triangle by rows, related to the numbers of binary trees of height less than n, derived from the Mandelbrot set.
%C A202019 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.
%C A202019 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:
%C A202019 2.............1.............3......
%C A202019 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".
%C A202019 Row sums = A003095: (1, 2, 5, 26, 677,...) relating to the number of binary trees of height less than n.
%C A202019 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
%D A202019 Persi Diaconis & R. L. Graham, "Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks", Princeton University Press, 2012; pp. 73-83.
%H A202019 Alois P. Heinz, <a href="/A202019/b202019.txt">Rows n = 1..13, flattened</a>
%H A202019 Juan Carlos Nuño and Francisco J. Muñoz, <a href="https://arxiv.org/abs/2112.15563">Entropy-Variance curves of binary sequences generated by random substitutions of constant length</a>, arXiv:2112.15563 [math.PR], 2021.
%F A202019 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 + 2*x^3 + x^2 + x)...
%F A202019 T(n,k) = A309049(2^(n-1)-1,k-1). - _Alois P. Heinz_, Jul 11 2019
%e A202019 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.
%e A202019 Triangle begins:
%e A202019   1;
%e A202019   1, 1;
%e A202019   1, 2,  1,  1;
%e A202019   1, 4,  6,  6,  5,   2,   1,  1;
%e A202019   1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1;
%e A202019   ...
%p A202019 b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
%p A202019       x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
%p A202019     end:
%p A202019 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2^(n-1)-1)):
%p A202019 seq(T(n), n=1..7);  # _Alois P. Heinz_, Jul 11 2019
%t A202019 b[n_] := b[n] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f]*
%t A202019      b[n-1-f]]][Min[g-1, n-g/2]]][2^(Length[IntegerDigits[n, 2]]-1)]];
%t A202019 T[n_] := CoefficientList[b[2^(n-1)-1], x];
%t A202019 Array[T, 7] // Flatten (* _Jean-François Alcover_, Feb 19 2021, after _Alois P. Heinz_ *)
%Y A202019 Row sums are A003095.
%Y A202019 Cf. A137560 (reversed rows).
%Y A202019 Cf. A309049.
%K A202019 nonn,tabf
%O A202019 1,5
%A A202019 _Gary W. Adamson_, Dec 08 2011