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

1, 1, 2, 4, 8, 18, 43, 102, 251, 625, 1584, 4055, 10509, 27451, 72307, 191697, 511335, 1370995, 3693452, 9991671, 27133149, 73934800, 202096673, 553999573, 1522651908, 4195087022, 11583820212, 32052475655, 88860186023

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

Index entries for sequences related to rooted trees

Formula

a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(4)=1, b(k)=0 if k != 4.

a(n) ~ c * d^n / n^(3/2), where d = 2.9224691962496551739365155005926289..., c = 0.43112017460637374030857983498164... . - 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=4, 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[DivisorSum[j, #*(g[# - 1] - If[# == 4, 1, 0])&] * g[n - j], {j, 1, n}]/n];
a[n_] := g[n - 1];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Apr 04 2017, after Alois P. Heinz *)

Crossrefs

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

Column k=4 of A255636.

Sequence in context: A049075 A318797 A318850 * A059221 A193617 A233139

Adjacent sequences: A052324 A052325 A052326 * A052328 A052329 A052330

Keyword

nonn

Author

Christian G. Bower, Dec 15 1999

Status

approved