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A052327 Number of rooted trees with a forbidden limb of length 4.

%I A052327 #19 Apr 04 2017 08:06:40
%S A052327 1,1,2,4,8,18,43,102,251,625,1584,4055,10509,27451,72307,191697,
%T A052327 511335,1370995,3693452,9991671,27133149,73934800,202096673,553999573,
%U A052327 1522651908,4195087022,11583820212,32052475655,88860186023
%N A052327 Number of rooted trees with a forbidden limb of length 4.
%C A052327 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 A052327 Alois P. Heinz, <a href="/A052327/b052327.txt">Table of n, a(n) for n = 1..1000</a>
%H A052327 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A052327 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052327 a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(4)=1, b(k)=0 if k != 4.
%F A052327 a(n) ~ c * d^n / n^(3/2), where d = 2.9224691962496551739365155005926289..., c = 0.43112017460637374030857983498164... . - _Vaclav Kotesovec_, Aug 25 2014
%p A052327 with(numtheory):
%p A052327 g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
%p A052327       `if`(d=4, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
%p A052327     end:
%p A052327 a:= n-> g(n-1):
%p A052327 seq(a(n), n=1..35);  # _Alois P. Heinz_, Jul 04 2014
%t A052327 g[n_] := g[n] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[# - 1] - If[# == 4, 1, 0])&] * g[n - j], {j, 1, n}]/n];
%t A052327 a[n_] := g[n - 1];
%t A052327 Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Apr 04 2017, after _Alois P. Heinz_ *)
%Y A052327 Cf. A002955, A002988-A002992, A052318-A052329.
%Y A052327 Column k=4 of A255636.
%K A052327 nonn
%O A052327 1,3
%A A052327 _Christian G. Bower_, Dec 15 1999