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

%I A052788 #41 Jan 13 2024 20:32:33
%S A052788 0,1,5,40,360,3570,37476,410490,4635330,53589045,631115140,7544876956,
%T A052788 91321148575,1116879203135,13781214640630,171350293212360,
%U A052788 2144719821588471,27001925967762160,341717698703959875
%N A052788 Number of rooted trees with n nodes and 5-colored non-root nodes.
%C A052788 Previous name was: A simple grammar.
%C A052788 Number of rooted trees with 5-colored non-root nodes. - _Christian G. Bower_, Sep 07 2002
%H A052788 Vaclav Kotesovec, <a href="/A052788/b052788.txt">Table of n, a(n) for n = 0..870</a>
%H A052788 L. Foissy, <a href="https://arxiv.org/abs/1811.07572">Algebraic structures on typed decorated rooted trees</a>, arXiv:1811.07572 [math.RA], 2018-2021.
%H A052788 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=745">Encyclopedia of Combinatorial Structures 745</a>.
%H A052788 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052788 a(n) ~ c * d^n / n^(3/2), where d = 13.78565111008468519893032491082181549507446564..., c = 0.0809706405011433830276324977466118885837... . - _Vaclav Kotesovec_, Aug 26 2014
%F A052788 G.f. A(x) satisfies: A(x) = x*exp(5*Sum_{k>=1} A(x^k)/k). - _Ilya Gutkovskiy_, Mar 19 2018
%p A052788 spec := [S,{B=Set(S),S=Prod(Z,B,B,B,B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052788 with(numtheory):
%p A052788 a:= proc(n) option remember; `if`(n<2, n, (add(add(d*
%p A052788       a(d), d=divisors(j))*a(n-j)*5, j=1..n-1))/(n-1))
%p A052788     end:
%p A052788 seq(a(n), n=0..25); # _Vaclav Kotesovec_, Aug 26 2014 after Alois P. Heinz
%t A052788 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 *)
%Y A052788 Column k=5 of A242249.
%K A052788 easy,nonn,eigen
%O A052788 0,3
%A A052788 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052788 New name from _Vaclav Kotesovec_, Aug 26 2014