A347928 - OEIS

%I #24 Jul 07 2022 05:16:56

%S 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,

%T 42,0,0,0,640,800,720,504,132,0,2048,1792,2688,3920,3584,3080,1848,

%U 429,0,0,4096,4608,11520,17760,16512,13104,6864,1430

%N 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.

%C 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.

%H Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, <a href="https://doi.org/10.5206/mt.v1i1.14037">Some Facts and Conjectures about Mandelbrot Polynomials</a>, Maple Trans., Vol. 1, No. 1, Article 14037 (July 2021).

%H V. P. Johnson, <a href="http://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. Southern Calif., 2012.

%H Michael Larsen, <a href="https://doi.org/10.1090/mcom/3564">Multiplicative series, modular forms, and Mandelbrot polynomials</a>, in: Mathematics of Computation 90.327 (Sept. 2020), pp. 345-377. Preprint: <a href="https://arxiv.org/abs/1908.09974">arXiv:1908.09974</a> [math.NT], 2019.

%F The Mandelbrot-Larsen polynomials are defined: M(0, x) = 0; M(1, x) = x/2;

%F 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.

%F P(n, x) = 2^(2*n-1)*M(n, x) (scaled Mandelbrot-Larsen polynomials).

%F T(n, k) = [x^k] P(n, x).

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

%F M(n, 2*k) = P(n, 2*k) / 2^(2*n-1) = A319539(n, k).

%F P(n, k) = A348686(n, k).

%F T(n, n) = A000108(n-1) for n >= 1, Catalan numbers.

%F T(n+2, n+1) / 2 = A000984(n) for n >= 0, central binomials.

%F P(n, 1) = A088674(n-1) for n >= 1, also row sums.

%F M(n, 2) = A001190(n) for n >= 0.

%F M(n, 4) = A083563(n) for n >= 0.

%F M(n,-4) = -A107087(n) for n >= 1.

%F M(n, 6) = A220816(n) for n >= 1.

%F M(n, 8) = A220817(n) for n >= 1.

%F Conjecture (Calkin, Chan, & Corless): content(P(n)) = gcd(row(n)) = A048896(n-1) for n >= 1.

%e Triangle starts:

%e [0]  0;

%e [1]  0,    1;

%e [2]  0,    2,    1;

%e [3]  0,    0,    4,    2;

%e [4]  0,   16,   12,   12,     5;

%e [5]  0,    0,   32,   40,    40,    14;

%e [6]  0,    0,  192,  208,   168,   140,    42;

%e [7]  0,    0,    0,  640,   800,   720,   504,   132;

%e [8]  0, 2048, 1792, 2688,  3920,  3584,  3080,  1848,  429;

%e [9]  0,    0, 4096, 4608, 11520, 17760, 16512, 13104, 6864, 1430.

%p M := proc(n, x) local k; option remember;

%p if n = 0 then 0 elif n = 1 then x else add(M(k, x)*M(n-k, x), k = 1..n-1) +

%p ifelse(n::even, M(n/2, x), 0) fi; expand(%/2) end:

%p P := n -> 2^(2*n - 1)*M(n, x):

%p row := n -> seq(coeff(P(n), x, k), k = 0..n): seq(row(n), n = 0..9);

%t 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]];

%t P[n_] := 2^(2*n - 1)*M[n, x];

%t row [n_] := If[n == 0, {0}, CoefficientList[P[n], x]];

%t Table[row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Jul 07 2022, after Maple code *)

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

%Y Cf. A319539, A348678, A348679, A348686.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Oct 27 2021