A002990 Number of n-node trees with a forbidden limb of length 4. (Formerly M0350)

1, 1, 1, 1, 2, 2, 5, 9, 19, 38, 86, 188, 439, 1026, 2472, 5997, 14835, 36964, 93246, 236922, 607111, 1565478, 4062797, 10599853, 27797420, 73224806, 193709710, 514406793, 1370937140, 3665714528, 9831891555, 26445886506, 71325268179

Offset

0,5

Comments

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

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

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 A052327.

a(n) ~ c * d^n / n^(5/2), where d = 2.9224691962496551739365155005926..., c = 0.503471518908815272581177797536... . - Vaclav Kotesovec, Aug 25 2014

Maple

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=4, 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 == 4, 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 26 2015, after Alois P. Heinz *)

Crossrefs

Cf. A002955, A002988, A002989, A002990, A002991, A002992.

Cf. A052318, A052319, A052320.

Cf. A052321, A052322, A052323, A052324, A052325, A052326.

Cf. A052327, A052328, A052329.

Sequence in context: A214727 A302483 A052969 * A060405 A326493 A003228

Adjacent sequences: A002987 A002988 A002989 * A002991 A002992 A002993

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved