A002989 Number of n-node trees with a forbidden limb of length 3. (Formerly M1082)

1, 1, 1, 1, 1, 2, 4, 7, 14, 28, 61, 131, 297, 678, 1592, 3770, 9096, 22121, 54451, 135021, 337651, 849698, 2152048, 5479408, 14022947, 36048514, 93061268, 241160180, 627179689, 1636448181, 4282964600, 11241488853, 29584389474

Offset

0,6

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 A052321.

a(n) ~ c * d^n / n^(5/2), where d = 2.851157026715821487965080545784..., c = 0.463162985533004672966744142107... . - 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=3, 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 == 3, 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-A002992, A052318-A052329.

Sequence in context: A119341 A119342 A119268 * A293336 A321401 A000671

Adjacent sequences: A002986 A002987 A002988 * A002990 A002991 A002992

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved