A052524 Number of ordered labeled rooted trees on n nodes with non-leaf nodes having more than two children.

0, 1, 0, 6, 24, 480, 5760, 126000, 2580480, 69310080, 1959552000, 64505548800, 2292022656000, 90366525849600, 3843167789260800, 177248722210560000, 8758468152225792000, 463225965106544640000, 26058454876652470272000

Offset: 0

Author

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

Keywords

• easy
• nonn

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

Crossrefs

Cf. A046736.

Programs

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 *)

PARI

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

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