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A052524 Number of ordered labeled rooted trees on n nodes with non-leaf nodes having more than two children. 3
0, 1, 0, 6, 24, 480, 5760, 126000, 2580480, 69310080, 1959552000, 64505548800, 2292022656000, 90366525849600, 3843167789260800, 177248722210560000, 8758468152225792000, 463225965106544640000, 26058454876652470272000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The correspondence between rooted trees and dissection of (n+1)-gon as in A046736 is just like the case for Catalan numbers and binary trees.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..370

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 94

FORMULA

a(n) = n! * A046736(n+1) for n>0.

E.g.f.: A(x)=sum_{n>0} a(n)*x^n/n! satisfies A(x)-A(x)^2-A(x)^3 = x*(1-A(x)).

Recurrence: a(0)=0, a(1)=1, a(2)=0, a(3)=6, 8*n*(n+1)*(n+2)*(1-2*n)*a(n) +6*(13*n+10)*(2*n+1)*(n+2)*a(n+1) -24*(2*n+5)*(4*n+7)*a(n+2) -4*(19*n+40)*a(n+3) +35*a(n+4) = 0

a(n) ~ n^(n-1) * sqrt(r*(1-s)/(2+6*s)) / (exp(n) * r^n), where r = 0.2933671276754004454... is the root of the equation 5-8*r-32*r^2+4*r^3 = 0 and s = 0.40303171676268477587... is the root of the equation 1-2*s-2*s^2+2*s^3 = 0. - Vaclav Kotesovec, Jan 08 2014

MAPLE

spec := [S, {S=Union(Z, Sequence(S, card >= 3))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

MATHEMATICA

CoefficientList[InverseSeries[Series[1 + 1/(x-1) + 2*x + x^2, {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 08 2014 *)

PROG

(PARI) a(n)=if(n<1, 0, n!*polcoeff(serreverse((x-x^2-x^3)/(1-x) + O(x^(n+2))), n))

CROSSREFS

Cf. A046736.

Sequence in context: A128614 A285018 A139240 * A267032 A234635 A036284

Adjacent sequences:  A052521 A052522 A052523 * A052525 A052526 A052527

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

STATUS

approved

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Last modified June 24 01:55 EDT 2021. Contains 345404 sequences. (Running on oeis4.)