%I A023431
%S A023431 1,1,1,2,4,7,13,26,52,104,212,438,910,1903,4009,8494,18080,38656,82988,
%T A023431 178802,386490,837928,1821664,3970282,8673258,18987930,41652382,
%U A023431 91539466,201525238,444379907,981384125,2170416738,4806513660
%N A023431 Generalized Catalan Numbers.
%C A023431 Essentially the same as A025246.
%C A023431 Number of lattice paths in the first quadrant from (0,0) to (n,0) using 
               only steps H=(1,0), U=(1,1) and D=(2,-1). E.g. a(5)=7 because we 
               have HHHHH, HHUD, HUDH, HUHD, UDHH, UHDH and UHHD. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Dec 25 2003
%C A023431 Also number of peakless Motzkin paths of length n with no double rises; 
               in other words, Motzkin paths of length n with no UD's and no UU's, 
               where U=(1,1) and D=(1,-1). E.g. a(5)=7 because we have HHHHH, HHUHD, 
               HUHDH, HUHHD, UHDHH, UHHDH and UHHHD, where H=(1,0). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Jan 09 2004
%C A023431 Series reversion of g.f. A(x) is -A(-x) (if offset 1). - Michael Somos, 
               Jul 13 2003
%C A023431 Hankel transform is A010892(n+1). [From Paul Barry (pbarry(AT)wit.ie), 
               Sep 19 2008]
%H A023431 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=666">
               Encyclopedia of Combinatorial Structures 666</a>
%F A023431 G.f.: (1-x-sqrt((1-x)^2-4x^3))/(2x^3)=A(x). y=xA(x) satisfies x-y+xy+(xy)^2=0.
%F A023431 a(n+1)=a(n)+a(0)a(n-2)+a(1)a(n-3)+...+a(n-2)a(0).
%F A023431 G.f.: (1/(1-x))c(x^3/(1-x)^2), c(x) the g.f. of A000108. [From Paul Barry 
               (pbarry(AT)wit.ie), Sep 19 2008]
%F A023431 Contribution from Paul Barry (pbarry(AT)wit.ie), May 22 2009: (Start)
%F A023431 G.f.: 1/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-... (continued fraction).
%F A023431 a(n)=sum{k=0..floor(n/3), C(n-k,2k)*A000108(k)}. (End)
%t A023431 Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ 
               n-3-k ], {k, 0, n-3} ];
%o A023431 (PARI) a(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^3+x^4*O(x^n)))/2,n+3)
%Y A023431 Cf. A000108, A001006, A004148, A006318.
%Y A023431 A025246(n+3)=a(n).
%Y A023431 Sequence in context: A017995 A099155 A068031 this_sequence A025246 A112740 
               A136408
%Y A023431 Adjacent sequences: A023428 A023429 A023430 this_sequence A023432 A023433 
               A023434
%K A023431 nonn,easy
%O A023431 0,4
%A A023431 Olivier Gerard (olivier.gerard(AT)gmail.com)