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%I M4274 #34 Jan 31 2015 11:52:21
%S 1,6,65,1092,25272,749034,27108440,1159194472,57190952440,
%T 3197759266112,199831490658912,13802087001056704,1044075809166477232,
%U 85847947926743165952,7623428923066363040672,727116625218755662644416
%N Number of labeled M-type rooted trees on n nodes.
%D 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.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 210, 242 (3.2.68, 3.3.92)
%D G. Labelle, Some new computational methods in the theory of species, pp. 192-209 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f. satisfies 2*A(x) = exp(x+A(x)) - 1 - log(1-x)*A(x).
%F 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
%t 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. *)
%o (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 */
%Y Cf. A052315.
%K nonn,eigen,nice
%O 1,2
%A _Simon Plouffe_ and _N. J. A. Sloane_
%E More terms, formula from _Christian G. Bower_, Dec 15 1999
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