%I #22 Apr 18 2017 07:04:11

%S 1,1,2,3,6,10,20,38,79,162,348,751,1661,3695,8335,18919,43310,99664,

%T 230700,536327,1252298,2934508,6900046,16272771,38484608,91243851,

%U 216839820,516423974,1232376315,2946354649,7056335478,16926813939,40665822366,97836595223,235697108115,568531616730

%N A simple grammar.

%C a(n) is the number of forests of rooted unlabeled identity trees. Euler transform of A004111. - _Geoffrey Critzer_, Oct 13 2012

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 64, Eq. (3.3.15); p. 80, Problem 3.10.

%H Alois P. Heinz, <a href="/A052843/b052843.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=811">Encyclopedia of Combinatorial Structures 811</a>

%F O.g.f.: prod(n>=1, 1/(1 - x^n)^A004111(n) ). - _Geoffrey Critzer_, Oct 13 2012

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

%p # second Maple program:

%p with(numtheory):

%p b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(d*

%p b(d)*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))

%p end:

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d),

%p d=divisors(j)) *a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Mar 06 2015

%t nn=35;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,-s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);it=Table[a[i],{i,1,nn}];CoefficientList[Series[Product[1/(1-x^i)^it[[i]],{i,1,nn}],{x,0,nn}],x] (* after code given by _Robert A. Russell_ in A004111 *)

%t (* _Geoffrey Critzer_, Oct 13 2012 *)

%Y Cf. A004111.

%K easy,nonn

%O 0,3

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

%E Added more terms, _Geoffrey Critzer_, Oct 13 2012