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A002988
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Number of trimmed trees with n nodes.
(Formerly M0777)
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23
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1, 1, 1, 0, 1, 1, 2, 3, 6, 10, 21, 39, 82, 167, 360, 766, 1692, 3726, 8370, 18866, 43029, 98581, 227678, 528196, 1232541, 2888142, 6798293, 16061348, 38086682, 90607902, 216230205, 517482053, 1241778985, 2987268628, 7203242490
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OFFSET
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0,7
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COMMENTS
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A trimmed tree is a tree with a forbidden limb of length 2.
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. (End)
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REFERENCES
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K. L. McAvaney, personal communication.
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).
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(5/2), where d = 2.59952511060090659632378883695..., c = 0.3758284247032014502508501798... . - Vaclav Kotesovec, Aug 24 2014
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MAPLE
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with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
`if`(d=2, 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):
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MATHEMATICA
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g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 2, 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 25 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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