A002991 Number of n-node trees with a forbidden limb of length 5. (Formerly M0725)

1, 1, 1, 1, 2, 3, 5, 10, 21, 43, 97, 215, 503, 1187, 2876, 7033, 17510, 43961, 111664, 285809, 737632, 1915993, 5008652, 13163785, 34774873, 92282214, 245930746, 657931603, 1766481135, 4758553683, 12858286083, 34844908142, 94681272368

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. - Christian G. Bower, Dec 15 1999

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 A052328. - Christian G. Bower, Dec 15 1999

a(n) ~ c * d^n / n^(5/2), where d = 2.9447916575019743775137795109303..., c = 0.521642401804532770865780146005... . - 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=5, 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 == 5, 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: A132418 A024494 A131708 * A218532 A352879 A022861

Adjacent sequences: A002988 A002989 A002990 * A002992 A002993 A002994

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Dec 15 1999

Status

approved