%I A005809 M2995
%S A005809 1,3,15,84,495,3003,18564,116280,735471,4686825,30045015,193536720,
%T A005809 1251677700,8122425444,52860229080,344867425584,2254848913647,14771069086725,
%U A005809 96926348578605,636983969321700,4191844505805495,27619435402363035
%N A005809 Binomial coefficients C(3n,n).
%C A005809 Number of paths in Z x Z starting at (0,0) and ending at (3n,0) using 
               steps in {(1,1),(1,-2)}.
%C A005809 Number of even trees with 2n edges and one distinguished vertex. Even 
               trees are rooted plane trees where every vertex (includig root) has 
               even degree.
%C A005809 Hankel transform is 3^n*A051255(n), where A051255 is the Hankel transform 
               of C(3n,n)/(2n+1). - Paul Barry (pbarry(AT)wit.ie), Jan 21 2007
%D A005809 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005809 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 828.
%H A005809 T. D. Noe, <a href="b005809.txt">Table of n, a(n) for n=0..100</a>
%H A005809 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A005809 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A005809 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A005809 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">
               Random walks, trees and extensions of Riordan group techniques</a>
%F A005809 The g.f. R[ z_ ] below was found by Kurt Persson (kurt(AT)math.chalmers.se) 
               and communicated by Einar Steingrimsson (einar(AT)math.chalmers.se).
%F A005809 Using Stirling's formula in A000142 it easy to get the asymptotic expression 
               a(n) ~ 1/2 * (27/4)^n / sqrt(Pi*n / 3) - Dan Fux (dan.fux(AT)OpenGaia.com 
               or danfux(AT)OpenGaia.com), Apr 07 2001
%F A005809 a(n)=sum{k=0..n, C(n, k)C(2n, k) } - Paul Barry (pbarry(AT)wit.ie), May 
               15 2003
%F A005809 G.f. = 1/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e. (in 
               Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z); - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), May 22 2003
%F A005809 a(n) ~ 1/2*3^(1/2)*pi^(-1/2)*n^(-1/2)*2^(-2*n)*3^(3*n)*{1 - 7/72*n^-1 
               + 49/10368*n^-2 + 6425/2239488*n^-3 - ...} - Joe Keane (jgk(AT)jgk.org), 
               Nov 07 2003
%F A005809 a(n) = A006480(n)/A000984(n) - Lior Manor (lior.manor(AT)gmail.com) May 
               04 2004
%F A005809 a(n)=sum_{0<=i_1<=n, 0<= i_2<=n}binomial(n, i_1)*binomial(n, i_2)*binomial(n, 
               i_1+i_2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 14 2004
%F A005809 a(n)=sum{k=0..n, A109971(k)*3^k}; a(0)=1, a(n)=sum{k=0..n, 3^k*C(3n-k,
               n-k)2k/(3n-k)}, n>0; - Paul Barry (pbarry(AT)wit.ie), Jan 21 2007
%F A005809 a(n)=A085478(2n,n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 17 2009]
%t A005809 R[ z_ ] := ((2-18*z + 27*z^2 + 3^(3/2)*z^(3/2)*(27*z-4)^(1/2))/2)^(1/
               3); f[ z_ ] := ( (R[ z ])^3 + (1-3*z)*(R[ z ])^2 + (1-6*z)*R[ z ] 
               )/( (R[ z ])^4 + (1-6*z)*(R[ z ])^2 + (6*z-1)^2 )
%Y A005809 Sequence in context: A115910 A106569 A026032 this_sequence A067122 A093593 
               A011900
%Y A005809 Adjacent sequences: A005806 A005807 A005808 this_sequence A005810 A005811 
               A005812
%K A005809 nonn,easy,nice
%O A005809 0,2
%A A005809 N. J. A. Sloane (njas(AT)research.att.com).