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%I #37 Jun 07 2019 17:05:49
%S 0,0,0,0,12,120,750,3780,16856,69552,272250,1026300,3762132,13498056,
%T 47615750,165683700,570024240,1942538592,6566094450,22038141420,
%U 73510278380,243854707320,804962754750,2645408201700,8658857196552,28237920483600,91778694166250
%N Number of forests of labeled rooted trees with n nodes, containing exactly 2 trees of height one, all others having height zero.
%H Alois P. Heinz, <a href="/A133386/b133386.txt">Table of n, a(n) for n = 0..650</a>
%H A. P. Heinz, <a href="https://doi.org/10.1007/3-540-53504-7_68">Finding Two-Tree-Factor Elements of Tableau-Defined Monoids in Time O(n^3)</a>, Ed. S. G. Akl, F. Fiala, W. W. Koczkodaj: Advances in Computing and Information, ICCI90 Niagara Falls, LNCS 468, Springer-Verlag (1990), pp. 120-128.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (18,-141,630,-1767,3222,-3815,2826,-1188,216).
%F a(n) = n*(n-1) * Stirling2(n-1,3).
%F G.f.: -2*x^4*(85*x^4-180*x^3+141*x^2-48*x+6) / ((x-1)^3*(3*x-1)^3*(2*x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
%F a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=12, a(5)=120, a(6)=750, a(7)=3780, a(8)=16856, a(n)=18*a(n-1)-141*a(n-2)+630*a(n-3)-1767*a(n-4)+ 3222*a(n-5)- 3815*a(n-6)+2826*a(n-7)-1188*a(n-8)+216*a(n-9). - _Harvey P. Dale_, May 02 2015
%e a(4) = 12 because 12 trees of the given kind exist: 1<-3 2<-4, 1<-4 2<-3, 1<-2 3<-4, 1<-4 3<-2, 1<-2 4<-3, 1<-3 4<-2, 2<-1 3<-4, 2<-4 3<-1, 2<-1 4<-3, 2<-3 4<-1, 3<-1 4<-2 and 3<-2 4<-1.
%p a:= n-> n*(n-1)*Stirling2(n-1, 3):
%p seq(a(n), n=0..50);
%t Join[{0},Table[n(n-1)StirlingS2[n-1,3],{n,30}]] (* or *) LinearRecurrence[{18,-141,630,-1767,3222,-3815,2826,-1188,216},{0,0,0,0,12,120,750,3780,16856},30] (* _Harvey P. Dale_, May 02 2015 *)
%Y Cf. A058877, A000248.
%Y Column k=2 of A133399.
%Y Column 2 of A198204. - _Peter Bala_, Aug 01 2012
%K nonn
%O 0,5
%A _Alois P. Heinz_, Nov 22 2007
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