A083563 -id:A083563 - cp's OEIS frontend

A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.

1, 2, 1, 3, 3, 1, 4, 6, 6, 2, 5, 10, 18, 18, 3, 6, 15, 40, 75, 54, 6, 7, 21, 75, 215, 333, 183, 11, 8, 28, 126, 495, 1260, 1620, 636, 23, 9, 36, 196, 987, 3600, 8010, 8208, 2316, 46, 10, 45, 288, 1778, 8568, 28275, 53240, 43188, 8610, 98, 11, 55, 405, 2970, 17934, 80136, 232500, 366680, 232947, 32763, 207

Offset: 1

Andrew Howroyd, Sep 22 2018

Not all k colors need to be used. The total number of nodes will be 2n-1.

See table 2.1 in the Johnson reference.

			Array begins:
===========================================================
n\k|  1    2      3       4        5        6         7
---+-------------------------------------------------------
1  |  1    2      3       4        5        6         7 ...
2  |  1    3      6      10       15       21        28 ...
3  |  1    6     18      40       75      126       196 ...
4  |  2   18     75     215      495      987      1778 ...
5  |  3   54    333    1260     3600     8568     17934 ...
6  |  6  183   1620    8010    28275    80136    194628 ...
7  | 11  636   8208   53240   232500   785106   2213036 ...
8  | 23 2316  43188  366680  1979385  7960638  26037431 ...
9  | 46 8610 232947 2590420 17287050 82804806 314260765 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

Columns 1..5 are A001190, A083563, A220816, A220817, A220818.
Main diagonal is A319580.
Cf. A241555, A319254, A319541.

A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 23 2018

A[n_, k_] := A[n, k] = If[n<2, k*n, If[OddQ[n], 0, (#*(1-#)/2&)[A[n/2, k]]] + Sum[A[i, k]*A[n-i, k], {i, 1, n/2}]];
Table[A[n, d-n+1], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz *)

\\ here R(n,k) gives k-th column as a vector.
R(n,k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n,]))}

T(1,k) = k.
T(n,k) = (1/2)*([n mod 2 == 0]*T(n/2,k) + Sum_{j=1..n-1} T(j,k)*T(n-j,k)).
G.f. of k-th column R(x) satisfies R(k) = k*x + (R(x)^2 + R(x^2))/2.

A220826 Number of unrooted binary leaf-multi-labeled trees with n leaves on the label set [2].

Original entry on oeis.org

2, 3, 4, 6, 12, 31, 78, 234, 722, 2376, 8046, 28263, 101226, 370389, 1375728, 5182107, 19743204, 75994993, 295110996, 1155128397, 4553360558, 18063221619, 72069527418, 289053637621, 1164871141254, 4714973350560, 19161577759814, 78162897838290, 319940064689142

Offset: 1

N. J. A. Sloane, Dec 22 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Column 2 of A339649.

Cf. A083563, A220829.

U(30,2) \\ See A339649 for U(n,k). - Andrew Howroyd, Dec 14 2020

Terms a(11) and beyond from Andrew Howroyd, Dec 14 2020

A347928 Triangle read by rows, T(n, k) are the coefficients of the scaled Mandelbrot-Larsen polynomials P(n, x) = 2^(2n-1)M(n, x), where M(n, x) are the Mandelbrot-Larsen polynomials; for 0 <= k <= n.

Original entry on oeis.org

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

Peter Luschny, Oct 27 2021

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.

			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.

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.

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

Cf. A319539, A348678, A348679, A348686.

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);

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 *)

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.

A220819 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].

Original entry on oeis.org

0, 1, 4, 14, 48, 171, 614, 2270, 8518, 32567, 126168, 495079, 1962752, 7853581, 31672502, 128622480, 525523990, 2158818376, 8911039462, 36941520279, 153740822408, 642085403709, 2690217364606, 11304538078369, 47630350694248, 201181246749072, 851690546714230

Offset: 1

N. J. A. Sloane, Dec 22 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1542 (first 200 terms from Andrew Howroyd)
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Column k=2 of A319541.

Cf. A001190, A083563.

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> b(n, 2)-2*b(n, 1):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 07 2019

(* b = A083563 *) b[n_] := b[n] = If[n < 2, 2*n, If[OddQ[n], 0, #*(1 - #)/2 &[b[n/2]]]] + Sum[b[i]*b[n - i], {i, 1, n/2}];
(* c = A001190 *) c[n_?OddQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, (n - 1)/2}]; c[n_?EvenQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, n/2 - 1}] + (1/2)*c[n/2]*(1 + c[n/2]); c[0] = 0; c[1] = 1;
a[n_] := b[n] - 2 c[n];
Array[a, 27] (* Jean-François Alcover, Sep 07 2019 *)

a(n) = A083563(n) - 2*A001190(n). - Andrew Howroyd, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018

cp's OEIS Frontend

A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.

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A220826 Number of unrooted binary leaf-multi-labeled trees with n leaves on the label set [2].

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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.

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Maple

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A220819 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A347928 Triangle read by rows, T(n, k) are the coefficients of the scaled Mandelbrot-Larsen polynomials P(n, x) = 2^(2n-1)M(n, x), where M(n, x) are the Mandelbrot-Larsen polynomials; for 0 <= k <= n.