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A052321
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Number of rooted trees with a forbidden limb of length 3.
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5
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1, 1, 2, 3, 7, 15, 35, 81, 195, 473, 1171, 2924, 7396, 18848, 48446, 125311, 326145, 853188, 2242616, 5919197, 15683008, 41694334, 111195166, 297393668, 797475499, 2143631474, 5775002574, 15590201095, 42168292074, 114260967888, 310124721255, 843053354234
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OFFSET
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1,3
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COMMENTS
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A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.
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LINKS
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FORMULA
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a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(3)=1, b(k)=0 if k != 3.
a(n) ~ c * d^n / n^(3/2), where d = 2.851157026715821487965080545784048..., c = 0.4192933669718878505916053142459... . - 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=3, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
end:
a:= n-> g(n-1):
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MATHEMATICA
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g[n_] := g[n] = If[n==0, 1, Sum[DivisorSum[j, #*(g[#-1] - If[#==3, 1, 0])&] * g[n-j], {j, 1, n}]/n];
a[n_] := g[n-1];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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