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A002988 Trimmed trees with n nodes.
(Formerly M0777)
23
1, 1, 1, 0, 1, 1, 2, 3, 6, 10, 21, 39, 82, 167, 360, 766, 1692, 3726, 8370, 18866, 43029, 98581, 227678, 528196, 1232541, 2888142, 6798293, 16061348, 38086682, 90607902, 216230205, 517482053, 1241778985, 2987268628, 7203242490 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

A trimmed tree is a tree with a forbidden limb of length 2.

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

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 = 0..1000

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)

A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972

Index entries for sequences related to trees

FORMULA

G.f.: 1+B(x)+(B(x^2)-B(x)^2)/2 where B(x) is g.f. of A002955.

a(n) ~ c * d^n / n^(5/2), where d = 2.59952511060090659632378883695..., c = 0.3758284247032014502508501798... . - Vaclav Kotesovec, Aug 24 2014

MAPLE

with(numtheory):

g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-

      `if`(d=2, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)

    end:

a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,

         g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):

seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2014

MATHEMATICA

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 2, 1, 0]), {d, Divisors[j] }]*g[n-j], {j, 1, n}]/n]; a[n_] := If[n == 0, 1, g[n-1] + (If[Mod[n, 2] == 0, g[Quotient[n, 2]-1], 0] - Sum[g[i-1]*g[n-i-1], {i, 1, n-1}])/2]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Feb 25 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A002955, A002989-A002992, A052318-A052329.

Sequence in context: A324407 A032291 A063687 * A138347 A211180 A265582

Adjacent sequences:  A002985 A002986 A002987 * A002989 A002990 A002991

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

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Last modified November 21 09:14 EST 2019. Contains 329362 sequences. (Running on oeis4.)