OFFSET
1,3
COMMENTS
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
FORMULA
a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(5)=1, b(k)=0 if k != 5.
a(n) ~ c * d^n / n^(3/2), where d = 2.944791657501974377513779510930324..., c = 0.43624554592719796037836168844839... . - Vaclav Kotesovec, Aug 25 2014
MAPLE
with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
`if`(d=5, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
end:
a:= n-> g(n-1):
seq(a(n), n=1..35); # Alois P. Heinz, Jul 04 2014
MATHEMATICA
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];
a[n_] := g[n-1];
Array[a, 35] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Dec 15 1999
STATUS
approved