%I #19 Mar 01 2015 18:56:05

%S 1,1,2,4,9,20,47,113,281,706,1807,4671,12224,32247,85782,229683,

%T 618767,1675618,4559263,12457483,34168574,94040433,259637564,

%U 718892281,1995739380,5553867981,15490305017,43293762352,121235084565

%N Number of rooted trees with a forbidden limb of length 6.

%C 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 Alois P. Heinz, <a href="/A052329/b052329.txt">Table of n, a(n) for n = 1..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(6)=1, b(k)=0 if k != 6.

%F a(n) ~ c * d^n / n^(3/2), where d = 2.95209316333202396584501452688304..., c = 0.43842619727838455589811980703038... . - _Vaclav Kotesovec_, Aug 25 2014

%p with(numtheory):

%p g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-

%p `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)

%p end:

%p a:= n-> g(n-1):

%p seq(a(n), n=1..35); # _Alois P. Heinz_, Jul 04 2014

%t 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 Cf. A002955, A002988-A002992, A052318-A052328.

%Y Column k=6 of A255636.

%K nonn

%O 1,3

%A _Christian G. Bower_, Dec 15 1999