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A002991
Number of n-node trees with a forbidden limb of length 5.
(Formerly M0725)
3
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).
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
KEYWORD
nonn
EXTENSIONS
More terms from Christian G. Bower, Dec 15 1999
STATUS
approved