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A202019
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Triangle by rows, related to the numbers of binary trees of height less than n, derived from the Mandelbrot set.
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3
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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
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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Persi Diaconis & R. L. Graham, "Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks", Princeton University Press, 2012; pp. 73-83.
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LINKS
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FORMULA
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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)...
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EXAMPLE
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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;
...
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MAPLE
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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)):
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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