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A035087
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Number of labeled rooted polygonal cacti (Husimi graphs) with n nodes.
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3
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1, 0, 3, 12, 135, 1440, 20895, 342720, 6585705, 142430400, 3449279295, 92207808000, 2699909867655, 85900402748160, 2951318065570875, 108894519775641600, 4294542443185019025, 180277244225580902400, 8025792422657714379675, 377695544010698833920000
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OFFSET
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1,3
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
Harary and E. M. Palmer, Graphical Enumeration, p. 71
F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141
F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953
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LINKS
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FORMULA
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E.g.f. satisfies A(x)=x*exp(A(x)^2/(2-2*A(x))).
a(n) ~ (1-s)^2 * sqrt(2/(6-11*s+4*s^2)) * n^(n-1) / (s * exp(1 - s^2/(2*(1-s))))^n, where s = 0.5391888728108891165... is the root of the equation 2-4*s+s^3=0. - 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)^2/ (2-2*A(n-1))), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..30); # Alois P. Heinz, Aug 22 2008
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
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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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