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
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).
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 27 2021
STATUS
approved