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A052328 Number of rooted trees with a forbidden limb of length 5.

%I A052328 #19 Dec 18 2020 04:17:09
%S A052328 1,1,2,4,9,19,46,110,273,684,1747,4505,11763,30956,82153,219437,
%T A052328 589747,1593170,4324445,11787195,32251520,88548011,243877256,
%U A052328 673605521,1865445693,5178574184,14408195935,40170674295,112213616851
%N A052328 Number of rooted trees with a forbidden limb of length 5.
%C A052328 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.
%H A052328 Alois P. Heinz, <a href="/A052328/b052328.txt">Table of n, a(n) for n = 1..1000</a>
%H A052328 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A052328 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052328 a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(5)=1, b(k)=0 if k != 5.
%F A052328 a(n) ~ c * d^n / n^(3/2), where d = 2.944791657501974377513779510930324..., c = 0.43624554592719796037836168844839... . - _Vaclav Kotesovec_, Aug 25 2014
%p A052328 with(numtheory):
%p A052328 g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
%p A052328       `if`(d=5, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
%p A052328     end:
%p A052328 a:= n-> g(n-1):
%p A052328 seq(a(n), n=1..35);  # _Alois P. Heinz_, Jul 04 2014
%t A052328 g[n_] := g[n] = If[n==0, 1, Sum[Sum[d(g[d-1] - If[d==5, 1, 0]), {d, Divisors[j]}] g[n-j], {j, 1, n}]/n];
%t A052328 a[n_] := g[n-1];
%t A052328 Array[a, 35] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A052328 Cf. A002955, A002988-A002992, A052318-A052329.
%Y A052328 Column k=5 of A255636.
%K A052328 nonn
%O A052328 1,3
%A A052328 _Christian G. Bower_, Dec 15 1999