A052763 Number of rooted trees with n nodes and 4-colored non-root nodes.

0, 1, 4, 26, 188, 1499, 12628, 111064, 1006840, 9345761, 88371580, 848273424, 8244075700, 80959901281, 802137370804, 8008422811882, 80488941119484, 813703130213745, 8268866850613468, 84417609311862182, 865408913186449784, 8905028017997573696

Offset: 0

Author

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

Keywords

• easy
• nonn
• eigen

Comments

Previous name was: A simple grammar.

Number of rooted trees with 4-colored non-root nodes. (Christian G. Bower, Sep 07 2002)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..950
• Chaim Even-Zohar, Calvin Leng, Counting Small Permutation Patterns, arXiv:1911.01414 [cs.DS], 2019.
• L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 720
• Index entries for sequences related to rooted trees

Crossrefs

Column k=4 of A242249.

Programs

Maple

spec := [S,{B=Set(S),S=Prod(Z,B,B,B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      a(d), d=divisors(j))*a(n-j)*4, j=1..n-1))/(n-1))
    end:
seq(a(n), n=0..25); # Vaclav Kotesovec, Aug 26 2014 after Alois P. Heinz

Mathematica

a[n_] := a[n] = If[n<2, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j]*4, {j, 1, n-1}]/(n-1)]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, adapted from Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 11.0699628777593263124193026233177403862890348..., c = 0.1016234204063820357399566577477318256736416... . - Vaclav Kotesovec, Aug 26 2014

G.f. A(x) satisfies: A(x) = x*exp(4*Sum_{k>=1} A(x^k)/k). - Ilya Gutkovskiy, Mar 19 2018

Extensions

New name from Vaclav Kotesovec, Aug 26 2014