A052327 - OEIS

%I #19 Apr 04 2017 08:06:40

%S 1,1,2,4,8,18,43,102,251,625,1584,4055,10509,27451,72307,191697,

%T 511335,1370995,3693452,9991671,27133149,73934800,202096673,553999573,

%U 1522651908,4195087022,11583820212,32052475655,88860186023

%N Number of rooted trees with a forbidden limb of length 4.

%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="/A052327/b052327.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(4)=1, b(k)=0 if k != 4.

%F a(n) ~ c * d^n / n^(3/2), where d = 2.9224691962496551739365155005926289..., c = 0.43112017460637374030857983498164... . - _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=4, 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[DivisorSum[j, #*(g[# - 1] - If[# == 4, 1, 0])&] * g[n - j], {j, 1, n}]/n];

%t a[n_] := g[n - 1];

%t Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Apr 04 2017, after _Alois P. Heinz_ *)

%Y Cf. A002955, A002988-A002992, A052318-A052329.

%Y Column k=4 of A255636.

%K nonn

%O 1,3

%A _Christian G. Bower_, Dec 15 1999