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A052318
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Number of labeled rooted trimmed trees with n nodes.
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19
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1, 2, 3, 16, 145, 1536, 19579, 290816, 4942305, 94689280, 2020278931, 47523053568, 1222147737265, 34117226135552, 1027550555918475, 33213871550365696, 1146891651823112641, 42135941698113503232, 1641164216596258397347, 67550839668807638712320
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OFFSET
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1,2
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COMMENTS
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A rooted trimmed tree is a tree with a forbidden limb of length 2.
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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E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014
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MAPLE
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A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x, n), polynom) fi end: a:= n-> coeff(A(n+1), x, n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008
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MATHEMATICA
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Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)], {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
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PROG
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(Maxima) a(n):=sum((if mod(n-m, 2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!, m, 1, n); \\ Vladimir Kruchinin, Aug 07 2012
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CROSSREFS
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KEYWORD
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nonn,eigen
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AUTHOR
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STATUS
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approved
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