A038079 -id:A038079 - cp's OEIS frontend

A052757 Number of rooted identity trees with n nodes and 3-colored non-root nodes.

0, 1, 3, 12, 64, 363, 2214, 14043, 91857, 614676, 4189254, 28974915, 202870938, 1435094800, 10241197917, 73639001172, 533004547453, 3880381334415, 28395656513145, 208748382089131, 1540935621796941, 11417266889312313, 84880193073070819, 632976019285857201

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

			a(3) = 12:
o  o  o  o  o  o  o  o  o    o      o      o
|  |  |  |  |  |  |  |  |   / \    / \    / \
1  1  1  2  2  2  3  3  3  1   2  1   3  2   3
|  |  |  |  |  |  |  |  |
1  2  3  1  2  3  1  2  3  - _Alois P. Heinz_, Feb 24 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 713

Cf. A038079.

Column k=3 of A255517.

spec := [S,{S=Prod(B,B,B,Z),B=PowerSet(S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

a(n) ~ c * d^n / n^(3/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938424..., c = 0.12982932099206082951153936270704832022771078... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 13 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(3*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(3*a(n)). (End)

New name from Vaclav Kotesovec, Feb 24 2015

A038077 Number of rooted identity trees with 2-colored nodes.

Original entry on oeis.org

2, 4, 10, 36, 132, 532, 2222, 9584, 42248, 189776, 864830, 3989656, 18593424, 87413444, 414060796, 1974260912, 9467922870, 45638482068, 221001289714, 1074591477112, 5244497440468, 25681907242416, 126149132242490, 621386729646752, 3068748094479108

Offset: 1

Christian G. Bower, Jan 04 1999

Shifts left and halves under Weigh transform.

Alois P. Heinz, Table of n, a(n) for n = 1..500
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A004111, A038078, A038079, A038080, A246312.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=1, 2, 2*b((n-1)$2)):
seq(a(n), n=1..40); # Alois P. Heinz, May 20 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n==1, 2, 2*b[n-1, n-1]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

a(n) = 2 * A005753(n).

A038080 Number of identity trees with 3-colored nodes.

Original entry on oeis.org

1, 3, 3, 9, 39, 189, 981, 5490, 31674, 189954, 1170126, 7382745, 47494197, 310712808, 2061987642, 13855192866, 94113385437, 645424668666, 4464027720900, 31110200069511, 218292811705458, 1541172223659249

Offset: 0

Christian G. Bower, Jan 04 1999

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Index entries for sequences related to trees

Cf. A000220, A038077-A038079, A052757, A255517.

G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038079.

a(n) ~ c * d^n / n^(5/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938... and c = 0.3712461766927875417276388215355520756010680416348018056669... - Vaclav Kotesovec, Dec 26 2020

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A052757 Number of rooted identity trees with n nodes and 3-colored non-root nodes.

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A038077 Number of rooted identity trees with 2-colored nodes.

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A038080 Number of identity trees with 3-colored nodes.

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