A052788 Number of rooted trees with n nodes and 5-colored non-root nodes.

0, 1, 5, 40, 360, 3570, 37476, 410490, 4635330, 53589045, 631115140, 7544876956, 91321148575, 1116879203135, 13781214640630, 171350293212360, 2144719821588471, 27001925967762160, 341717698703959875

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 5-colored non-root nodes. - Christian G. Bower, Sep 07 2002

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..870
• L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018-2021.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 745.
• Index entries for sequences related to rooted trees

Crossrefs

Column k=5 of A242249.

Programs

Maple

spec := [S,{B=Set(S),S=Prod(Z,B,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)*5, 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]*5, {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 = 13.78565111008468519893032491082181549507446564..., c = 0.0809706405011433830276324977466118885837... . - Vaclav Kotesovec, Aug 26 2014

G.f. A(x) satisfies: A(x) = x*exp(5*Sum_{k>=1} A(x^k)/k). - Ilya Gutkovskiy, Mar 19 2018

Extensions

New name from Vaclav Kotesovec, Aug 26 2014