OFFSET
0,7
COMMENTS
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)
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
FORMULA
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
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-François Alcover, Feb 25 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Christian G. Bower, Dec 15 1999
STATUS
approved