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A052329
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Number of rooted trees with a forbidden limb of length 6.
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19
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1, 1, 2, 4, 9, 20, 47, 113, 281, 706, 1807, 4671, 12224, 32247, 85782, 229683, 618767, 1675618, 4559263, 12457483, 34168574, 94040433, 259637564, 718892281, 1995739380, 5553867981, 15490305017, 43293762352, 121235084565
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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(6)=1, b(k)=0 if k != 6.
a(n) ~ c * d^n / n^(3/2), where d = 2.95209316333202396584501452688304..., c = 0.43842619727838455589811980703038... . - Vaclav Kotesovec, Aug 25 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=6, 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[Sum[d*(g[d-1]-If[d == 6, 1, 0]), {d, Divisors[j]} ]*g[n-j], {j, 1, n}]/n]; a[n_] := g[n-1]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)
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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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