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A052321 Number of rooted trees with a forbidden limb of length 3.

%I A052321 #27 Mar 05 2019 18:59:30
%S A052321 1,1,2,3,7,15,35,81,195,473,1171,2924,7396,18848,48446,125311,326145,
%T A052321 853188,2242616,5919197,15683008,41694334,111195166,297393668,
%U A052321 797475499,2143631474,5775002574,15590201095,42168292074,114260967888,310124721255,843053354234
%N A052321 Number of rooted trees with a forbidden limb of length 3.
%C A052321 A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.
%C A052321 Likely a duplicate of A003006. - _R. J. Mathar_, Mar 23 2012
%C A052321 Only first 10 terms match, but then a(11) = 1171, and A003006(11) = 1170. - _Vladimir Reshetnikov_, Mar 05 2019
%H A052321 Alois P. Heinz, <a href="/A052321/b052321.txt">Table of n, a(n) for n = 1..1000</a>
%H A052321 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A052321 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052321 a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(3)=1, b(k)=0 if k != 3.
%F A052321 a(n) ~ c * d^n / n^(3/2), where d = 2.851157026715821487965080545784048..., c = 0.4192933669718878505916053142459... . - _Vaclav Kotesovec_, Aug 24 2014
%p A052321 with(numtheory):
%p A052321 g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
%p A052321       `if`(d=3, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
%p A052321     end:
%p A052321 a:= n-> g(n-1):
%p A052321 seq(a(n), n=1..35);  # _Alois P. Heinz_, Jun 26 2014
%t A052321 g[n_] := g[n] = If[n==0, 1, Sum[DivisorSum[j, #*(g[#-1] - If[#==3, 1, 0])&] * g[n-j], {j, 1, n}]/n];
%t A052321 a[n_] := g[n-1];
%t A052321 Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Apr 04 2017, after _Alois P. Heinz_ *)
%Y A052321 Cf. A002955, A002988-A002992, A003006 (first 10 terms match), A052318-A052329.
%Y A052321 Column k=3 of A255636.
%K A052321 nonn
%O A052321 1,3
%A A052321 _Christian G. Bower_, Dec 15 1999