%I A047891
%S A047891 1,3,12,57,300,1686,9912,60213,374988,2381322,15361896,100389306,
%T A047891 663180024,4421490924,29712558576,201046204173,1368578002188,
%U A047891 9366084668802,64403308499592,444739795023054,3082969991029800
%N A047891 Number of planar rooted trees with n nodes and tricolored end nodes.
%C A047891 Also number of lattice paths from (0,0) to (n,n), with steps (1,0),(0,
               1) and (1,1), that never rise above the line y=x and the steps (1,
               1) are colored red or blue. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 28 2003
%C A047891 The Hankel transform (see A001906 for definition) of this sequence forms 
               A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ... ] . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2006
%C A047891 With a(0)=0, this is the series reversion of x(1-x)/(1+2x). [From Paul 
               Barry (pbarry(AT)wit.ie), Oct 18 2009]
%D A047891 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal 
               Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H A047891 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A047891 (1-2z-sqrt(4z^2-8z+1))/2z.
%F A047891 For n>0, a(n)=(1/n)*sum(k=0, n, 3^k*C(n, k)*C(n, k-1)) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), May 10 2003
%F A047891 a(1)=1, a(n)=2*a(n-1)+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 16 2004
%F A047891 The Hankel transform (see A001906 for definition) of this sequence form 
               A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ... ] . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2006
%F A047891 2*a(n)=A054872(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 
               17 2007
%F A047891 Contribution from Paul Barry (pbarry(AT)wit.ie), Feb 01 2009: (Start)
%F A047891 G.f.: 1/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);
%F A047891 a(n)=sum{k=0..n, C(n+k,2k)*2^(n-k)*A000108(k)}. (End)
%F A047891 G.f.: 1/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). 
               [From Paul Barry (pbarry(AT)wit.ie), Oct 18 2009]
%F A047891 a(0) = 1, for n>=1, a(n) = 3*A007564(n) [From Aoife Hennessy (aoife.hennessy(AT)gmail.com), 
               Dec 02 2009]
%o A047891 (PARI) a(n)=if(n<1,1,sum(k=0,n,3^k*binomial(n,k)*binomial(n,k-1))/n)
%Y A047891 Essentially the same as A025231.
%Y A047891 Cf. A006318.
%Y A047891 Sequence in context: A101106 A165310 A133158 this_sequence A151498 A103370 
               A094149
%Y A047891 Adjacent sequences: A047888 A047889 A047890 this_sequence A047892 A047893 
               A047894
%K A047891 nonn,eigen,easy,new
%O A047891 1,2
%A A047891 Louis Shapiro (lshapiro(AT)howard.edu)
%E A047891 More terms from Christian Bower, Dec 11 1999