A347928 Triangle read by rows, T(n, k) are the coefficients of the scaled Mandelbrot-Larsen polynomials P(n, x) = 2^(2*n-1)*M(n, x), where M(n, x) are the Mandelbrot-Larsen polynomials; for 0 <= k <= n.
0, 0, 1, 0, 2, 1, 0, 0, 4, 2, 0, 16, 12, 12, 5, 0, 0, 32, 40, 40, 14, 0, 0, 192, 208, 168, 140, 42, 0, 0, 0, 640, 800, 720, 504, 132, 0, 2048, 1792, 2688, 3920, 3584, 3080, 1848, 429, 0, 0, 4096, 4608, 11520, 17760, 16512, 13104, 6864, 1430
Offset: 0
Examples
Triangle starts: [0] 0; [1] 0, 1; [2] 0, 2, 1; [3] 0, 0, 4, 2; [4] 0, 16, 12, 12, 5; [5] 0, 0, 32, 40, 40, 14; [6] 0, 0, 192, 208, 168, 140, 42; [7] 0, 0, 0, 640, 800, 720, 504, 132; [8] 0, 2048, 1792, 2688, 3920, 3584, 3080, 1848, 429; [9] 0, 0, 4096, 4608, 11520, 17760, 16512, 13104, 6864, 1430.
Links
- Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Trans., Vol. 1, No. 1, Article 14037 (July 2021).
- V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.
- Michael Larsen, Multiplicative series, modular forms, and Mandelbrot polynomials, in: Mathematics of Computation 90.327 (Sept. 2020), pp. 345-377. Preprint: arXiv:1908.09974 [math.NT], 2019.
Crossrefs
Programs
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Maple
M := proc(n, x) local k; option remember; if n = 0 then 0 elif n = 1 then x else add(M(k, x)*M(n-k, x), k = 1..n-1) + ifelse(n::even, M(n/2, x), 0) fi; expand(%/2) end: P := n -> 2^(2*n - 1)*M(n, x): row := n -> seq(coeff(P(n), x, k), k = 0..n): seq(row(n), n = 0..9);
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Mathematica
M[n_, x_] := M[n, x] = Module[{k, w}, w = Which[n == 0, 0, n == 1, x, True, Sum[M[k, x]*M[n-k, x], {k, 1, n-1}] + If[EvenQ[n], M[n/2, x], 0]]; Expand[w/2]]; P[n_] := 2^(2*n - 1)*M[n, x]; row [n_] := If[n == 0, {0}, CoefficientList[P[n], x]]; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Jul 07 2022, after Maple code *)
Formula
The Mandelbrot-Larsen polynomials are defined: M(0, x) = 0; M(1, x) = x/2;
M(n, x) = (1/2)*(even(n)*M(n/2, x) + Sum_{k=1..n-1} M(k, x)*M(n-k, x)) for n > 1. Here even(n) = 1 if n is even, otherwise 0.
P(n, x) = 2^(2*n-1)*M(n, x) (scaled Mandelbrot-Larsen polynomials).
T(n, k) = [x^k] P(n, x).
M(n, 2*k) = P(n, 2*k) / 2^(2*n-1) = A319539(n, k).
P(n, k) = A348686(n, k).
T(n, n) = A000108(n-1) for n >= 1, Catalan numbers.
T(n+2, n+1) / 2 = A000984(n) for n >= 0, central binomials.
P(n, 1) = A088674(n-1) for n >= 1, also row sums.
M(n, 2) = A001190(n) for n >= 0.
M(n, 4) = A083563(n) for n >= 0.
M(n,-4) = -A107087(n) for n >= 1.
M(n, 6) = A220816(n) for n >= 1.
M(n, 8) = A220817(n) for n >= 1.
Conjecture (Calkin, Chan, & Corless): content(P(n)) = gcd(row(n)) = A048896(n-1) for n >= 1.
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