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A052329 Number of rooted trees with a forbidden limb of length 6.

%I A052329 #19 Mar 01 2015 18:56:05
%S A052329 1,1,2,4,9,20,47,113,281,706,1807,4671,12224,32247,85782,229683,
%T A052329 618767,1675618,4559263,12457483,34168574,94040433,259637564,
%U A052329 718892281,1995739380,5553867981,15490305017,43293762352,121235084565
%N A052329 Number of rooted trees with a forbidden limb of length 6.
%C A052329 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 A052329 Alois P. Heinz, <a href="/A052329/b052329.txt">Table of n, a(n) for n = 1..1000</a>
%H A052329 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A052329 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052329 a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(6)=1, b(k)=0 if k != 6.
%F A052329 a(n) ~ c * d^n / n^(3/2), where d = 2.95209316333202396584501452688304..., c = 0.43842619727838455589811980703038... . - _Vaclav Kotesovec_, Aug 25 2014
%p A052329 with(numtheory):
%p A052329 g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
%p A052329       `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
%p A052329     end:
%p A052329 a:= n-> g(n-1):
%p A052329 seq(a(n), n=1..35);  # _Alois P. Heinz_, Jul 04 2014
%t A052329 g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 6, 1, 0]), {d, Divisors[j]} ]*g[n-j], {j, 1, n}]/n]; a[n_] := g[n-1]; Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Feb 24 2015, after _Alois P. Heinz_ *)
%Y A052329 Cf. A002955, A002988-A002992, A052318-A052328.
%Y A052329 Column k=6 of A255636.
%K A052329 nonn
%O A052329 1,3
%A A052329 _Christian G. Bower_, Dec 15 1999