OFFSET
1,4
COMMENTS
A rooted trimmed tree is a tree without limbs of length >= 2. Limbs are the paths from the leafs (towards the root) to the nearest branching point (with the root considered to be a branching point). [clarified by Joerg Arndt, Mar 03 2015]
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.
Also counts the unordered rooted trees without "x x" in the level sequence for the pre-order walk. The bijection transforms the two outmost nodes in all limbs of lengths >= 2 into V-shaped subtrees. - Joerg Arndt, Mar 03 2015
REFERENCES
K. L. McAvaney, personal communication.
A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000 (terms n = 1..300 from Vincenzo Librandi)
F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
N. J. A. Sloane, Transforms
FORMULA
a(n) satisfies a=SHIFT_RIGHT(EULER(a-b)) where b(2)=1, b(k)=0 if k != 2.
a(n) ~ c * d^n / n^(3/2), where d = 2.59952511060090659632378883695107..., c = 0.391083882871301267612387143401... . - Vaclav Kotesovec, Aug 24 2014
MAPLE
with(numtheory): a:= proc(n) option remember; local d, j, aa; aa:= n-> a(n)-`if`(n=2, 1, 0); if n<=1 then n else (add(d*aa(d), d=divisors(n-1)) +add(add(d*aa(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..32); # Alois P. Heinz, Sep 06 2008
MATHEMATICA
a[n_] := a[n] = (Total[ #*b[#]& /@ Divisors[n-1] ] + Sum[ Total[ #*b[#]& /@ Divisors[j] ]*a[n-j], {j, 1, n-2}]) / (n-1); a[1] = 1; b[n_] := a[n]; b[2] = 0; Table[ a[n], {n, 1, 32}](* Jean-François Alcover, Nov 18 2011, after Maple *)
CROSSREFS
KEYWORD
nonn,nice,eigen
AUTHOR
EXTENSIONS
More terms, formula and comments from Christian G. Bower, Dec 15 1999
STATUS
approved