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A006959
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Number of labeled M-type rooted trees on n nodes.
(Formerly M4274)
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1
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1, 6, 65, 1092, 25272, 749034, 27108440, 1159194472, 57190952440, 3197759266112, 199831490658912, 13802087001056704, 1044075809166477232, 85847947926743165952, 7623428923066363040672, 727116625218755662644416
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OFFSET
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1,2
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Théorie des espèces et Combinatoire des Structures Arborescentes, Publications du LACIM, Université du Québec à Montréal, 1994, p. 214.
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 210, 242 (3.2.68, 3.3.92)
G. Labelle, Some new computational methods in the theory of species, pp. 192-209 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
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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E.g.f. satisfies 2*A(x) = exp(x+A(x)) - 1 - log(1-x)*A(x).
a(n) ~ n^(n-1) * sqrt(1 + (1+log(1-r))/((1-r)*(2+log(1-r))^2)) / (exp(n) * r^(n-1/2)), where r = 0.1520268451233936874315... is the root of the equation 2 + log(1-r) = exp(1+r-1/(2+log(1-r))). - Vaclav Kotesovec, Jan 08 2014
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MATHEMATICA
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max = 16; f[x_] := -1/(2+Log[1-x]) - ProductLog[-E^(x - 1/(2+Log[1-x]))/(2+Log[1-x])]; Rest[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!](* Jean-François Alcover, Mar 07 2012, after e.g.f. *)
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, A = 0; for( k=1, n, A += x * O(x^k); A = truncate( exp( x + A) - 1 - A*(1 + log( 1 - x + A - A)) )); n! * polcoeff( A, n))} /* Michael Somos, Jun 07 2012 */
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CROSSREFS
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KEYWORD
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nonn,eigen,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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