cp's OEIS Frontend

A002992 Number of n-node trees with a forbidden limb of length 6.

%I A002992 M0778 #37 Oct 21 2023 01:07:28
%S A002992 1,1,1,1,2,3,6,10,22,45,102,226,531,1253,3044,7456,18604,46798,119133,
%T A002992 305567,790375,2057523,5390759,14200122,37598572,100005401,267131927,
%U A002992 716318650,1927758155,5205240762,14098580633,38296720823,104308468102,284822276099
%N A002992 Number of n-node trees with a forbidden limb of length 6.
%C A002992 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.
%D A002992 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.
%D A002992 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002992 Alois P. Heinz, <a href="/A002992/b002992.txt">Table of n, a(n) for n = 0..1000</a>
%H A002992 A. J. Schwenk, <a href="/A002988/a002988.pdf">Letter to N. J. A. Sloane, Aug 1972</a>
%H A002992 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A002992 G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is g.f. of A052329.
%F A002992 a(n) ~ c * d^n / n^(5/2), where d = 2.95209316333202396584501452688304..., c = 0.52950413787119576841378912289... . - _Vaclav Kotesovec_, Aug 25 2014
%p A002992 with(numtheory):
%p A002992 g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
%p A002992       `if`(d=6, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
%p A002992     end:
%p A002992 a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,
%p A002992          g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):
%p A002992 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 06 2014
%t A002992 g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 6, 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_ *)
%Y A002992 Cf. A002955, A002988-A002991, A052318-A052329.
%K A002992 nonn
%O A002992 0,5
%A A002992 _N. J. A. Sloane_
%E A002992 More terms, formula and comments from _Christian G. Bower_, Dec 15 1999