A002955 Number of (unordered, unlabeled) rooted trimmed trees with n nodes.

1, 1, 1, 2, 4, 8, 17, 36, 79, 175, 395, 899, 2074, 4818, 11291, 26626, 63184, 150691, 361141, 869057, 2099386, 5088769, 12373721, 30173307, 73771453, 180800699, 444101658, 1093104961, 2695730992, 6659914175, 16481146479, 40849449618

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• eigen

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
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A002988-A002992, A052318-A052329.

Column k=2 of A255636.

Programs

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 *)

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

Extensions

More terms, formula and comments from Christian G. Bower, Dec 15 1999