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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

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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