0,7

From Christian G. Bower, Dec 15 1999: (Start)

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. (End)

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).

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

G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is the g.f. of A002955. - Christian G. Bower, Dec 15 1999

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

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

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-François Alcover, Feb 25 2015, after Alois P. Heinz *)

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

Sequence in context: A032291 A063687 A359019 * A138347 A211180 A265582

Adjacent sequences: A002985 A002986 A002987 * A002989 A002990 A002991

nonn,nice

More terms from Christian G. Bower, Dec 15 1999

approved