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

0,5

Comments

To avoid confusion: the polynomials which are called 'Mandelbrot polynomials' by some authors are listed in A137560. The name 'Mandelbrot-Larsen' polynomials was introduced in Calkin, Chan, & Corless to distinguish them from the Mandelbrot polynomials.

Links

Table of n, a(n) for n=0..54.

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.

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).

[x^k] M(n,k) = A348679(n, k) / A348678(n, k).

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.

Example

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.

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);

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 *)

Crossrefs

Cf. A137560, A000108, A000984, A001190, A083563, A088674, A048896, A107087, A220816, A220817.

Cf. A319539, A348678, A348679, A348686.

Sequence in context: A204040 A325773 A220779 * A317554 A089975 A034366

Adjacent sequences: A347925 A347926 A347927 * A347929 A347930 A347931

Keyword

nonn,tabl

Author

Peter Luschny, Oct 27 2021

Status

approved