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

%I A052524 #15 May 13 2019 01:06:43
%S A052524 0,1,0,6,24,480,5760,126000,2580480,69310080,1959552000,64505548800,
%T A052524 2292022656000,90366525849600,3843167789260800,177248722210560000,
%U A052524 8758468152225792000,463225965106544640000,26058454876652470272000
%N A052524 Number of ordered labeled rooted trees on n nodes with non-leaf nodes having more than two children.
%C A052524 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.
%H A052524 G. C. Greubel, <a href="/A052524/b052524.txt">Table of n, a(n) for n = 0..370</a>
%H A052524 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=94">Encyclopedia of Combinatorial Structures 94</a>
%F A052524 a(n) = n! * A046736(n+1) for n>0.
%F A052524 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)).
%F A052524 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
%F A052524 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
%p A052524 spec := [S,{S=Union(Z,Sequence(S,card >= 3))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052524 CoefficientList[InverseSeries[Series[1 + 1/(x-1) + 2*x + x^2, {x,0,20}], x], x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o A052524 (PARI) a(n)=if(n<1,0,n!*polcoeff(serreverse((x-x^2-x^3)/(1-x) + O(x^(n+2))), n))
%Y A052524 Cf. A046736.
%K A052524 easy,nonn
%O A052524 0,4
%A A052524 encyclopedia(AT)pommard.inria.fr, Jan 25 2000