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A292668 Number of forests of exactly n (unlabeled) ordered rooted trees with a total of 2n non-root nodes.

%I A292668 #13 Dec 21 2020 07:16:44
%S A292668 1,2,8,28,105,384,1442,5388,20317,76712,290790,1104538,4205909,
%T A292668 16044994,61322356,234739140,899911685,3454630372,13278582906,
%U A292668 51098682962,196853475135,759139115962,2930340545406,11321631496180,43779660235746,169429224658130
%N A292668 Number of forests of exactly n (unlabeled) ordered rooted trees with a total of 2n non-root nodes.
%C A292668 Each tree has at least 1 non-root node.
%H A292668 Alois P. Heinz, <a href="/A292668/b292668.txt">Table of n, a(n) for n = 0..1666</a>
%H A292668 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A292668 G.f.: Product_{j>=1} 1/(1-x^j)^A000108(j+1).
%F A292668 a(n) = A275431(2n,n).
%F A292668 a(n) ~ c * 4^n / n^(3/2), where c = 49.48222899350915021666300344559315... - _Vaclav Kotesovec_, Sep 27 2017
%e A292668 : a(2) = 8: (2 trees in each forest having 4 non-root nodes)
%e A292668 :
%e A292668 : o o . o o .  o o .  o  o .  o  o . o  o  .   o   o .  o   o  .
%e A292668 : | | . | | .  | | . ( ) | . ( ) | . | ( ) .  /|\  | . ( ) ( ) .
%e A292668 : o o . o o .  o o . o o o . o o o . o o o . o o o o . o o o o .
%e A292668 : |   . | | . ( )  . |     .   |   . |     .         .         .
%e A292668 : o   . o o . o o  . o     .   o   . o     .         .         .
%e A292668 : |   .     .      .       .       .       .         .         .
%e A292668 : o   .     .      .       .       .       .         .         .
%e A292668 :
%p A292668 C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A292668 a:= proc(n) option remember; `if`(n=0, 1, add(add(C(d+1)
%p A292668       *d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A292668     end:
%p A292668 seq(a(n), n=0..30);
%t A292668 c[n_] := c[n] = Binomial[2n, n]/(n+1);
%t A292668 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[c[d+1] d, {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
%t A292668 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y A292668 Cf. A000108, A275431.
%K A292668 nonn
%O A292668 0,2
%A A292668 _Alois P. Heinz_, Sep 20 2017