%I A064062
%S A064062 1,1,3,13,67,381,2307,14589,95235,636925,4341763,30056445,210731011,
%T A064062 1493303293,10678370307,76957679613,558403682307,4075996839933,
%U A064062 29909606989827,220510631755773,1632599134961667,12133359132082173
%N A064062 Generalized Catalan numbers C(2; n).
%C A064062 a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference 
               (see A064094) for alpha=2, beta =1 (or alpha=1, beta=2).
%C A064062 a(n) = number of Dyck n-paths (A000108) in which each upstep (U) not 
               at ground level is colored red (R) or blue (B). For example, a(3)=3 
               counts URDD, UBDD, UDUD (D=downstep). - David Callan (callan(AT)stat.wisc.edu), 
               Mar 30 2007
%C A064062 The Hankel transform of this sequence is A002416 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 19 2007
%C A064062 The sequence a(n)/2^n, with g.f. 1/(1-xc(x)/2), has Hankel transform 
               1/2^n. - Paul Barry (pbarry(AT)wit.ie), Apr 14 2008
%D A064062 N. Bonichon, C. Gavoille And N. Hanusse. Canonical Decomposition of Outerplanar 
               Maps and Application to Enumeration, Coding and Generation. In Proceedings 
               of WG'03, volume 2880 of LNCS, pp. 81-92, 2003.
%H A064062 Alexander Burstein, Sergi Elizalde and Toufik Mansour, <a href="http:/
               /arXiv.org/abs/math.CO/0610234">Restricted Dumont permutations, Dyck 
               paths and noncrossing partitions</a>, arXiv math.CO/0610234.
%F A064062 G.f.: (1+2*x*c(2*x))/(1+x) = 1/(1-x*c(2*x)) with c(x) g.f. of Catalan 
               numbers A000108.
%F A064062 a(n)= A062992(n-1) = sum((n-m)*binomial(n-1+m, m)*(2^m)/n, m=0..n-1), 
               n >= 1, a(0) := 1.
%F A064062 a(n) = Sum{ k= 0...n, A059365(n, k)*2^(n-k) }. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jan 19 2004
%F A064062 G.f.: 1/(1-x/(1-2x/(1-2x/(1-2x/(1-.... =1/(1-x-2x^2/(1-4x-4x^2/(1-4x-4x^2/
               (1-.... (continued fractions). [From Paul Barry (pbarry(AT)wit.ie), 
               Jan 30 2009]
%t A064062 a[0]=1; a[1]=1; a[n_]/;n>=2 := a[n] = a[n-1] + Sum[(a[k] + a[k-1])a[n-k],
               {k,n-1}]; Table[a[n],{n,0,10}] [From David Callan (callan(AT)stat.wisc.edu), 
               Aug 27 2009]
%o A064062 (PARI) a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x),n)
%Y A064062 Sequence in context: A136784 A027277 A062992 this_sequence A114191 A107592 
               A028418
%Y A064062 Adjacent sequences: A064059 A064060 A064061 this_sequence A064063 A064064 
               A064065
%K A064062 nonn,easy
%O A064062 0,3
%A A064062 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 
               2001