A006959 Number of labeled M-type rooted trees on n nodes. (Formerly M4274)

1, 6, 65, 1092, 25272, 749034, 27108440, 1159194472, 57190952440, 3197759266112, 199831490658912, 13802087001056704, 1044075809166477232, 85847947926743165952, 7623428923066363040672, 727116625218755662644416

Offset

1,2

References

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).

Links

Table of n, a(n) for n=1..16.

Index entries for sequences related to rooted trees

Formula

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

Mathematica

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. *)

Prog

(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 */

Crossrefs

Cf. A052315.

Sequence in context: A243698 A217899 A349524 * A121017 A239998 A278841

Adjacent sequences: A006956 A006957 A006958 * A006960 A006961 A006962

Keyword

nonn,eigen,nice

Author

Simon Plouffe and N. J. A. Sloane

Extensions

More terms, formula from Christian G. Bower, Dec 15 1999

Status

approved