Search: id:A023426
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%I A023426
%S A023426 1,1,1,1,2,4,7,11,18,32,59,107,191,343,627,1159,2146,
%T A023426 3972,7373,13757,25781,48437,91165,171945,325096,
%U A023426 616066,1169667,2224355,4236728,8082374,15441719
%N A023426 Generalized Catalan Numbers.
%C A023426 Number of lattice paths from (0,0) to (n,0) that stay weakly in the first 
               quadrant and such that each step is either U=(2,1),D=(2,-1), or H=(1,
               0). E.g. a(5)=4 because we have HHHHH, HUD, UDH and UHD. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Dec 23 2003
%C A023426 Hankel transform is A132380(n+3). [From Paul Barry (pbarry(AT)wit.ie), 
               May 22 2009]
%F A023426 G.f.=[1-z-sqrt((1-z)^2-4z^4)]/[2z^4]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 23 2003
%F A023426 Contribution from Paul Barry (pbarry(AT)wit.ie), May 22 2009: (Start)
%F A023426 G.f.: 1/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-... (continued fraction).
%F A023426 G.f.: (1/(1-x))c(x^4/(1-x)^2), c(x) the g.f. of A000108.
%F A023426 a(n)=sum{k=0..floor(n/4), C(n-2k,2k)*A000108(k)}. (End)
%t A023426 Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ 
               n-4-k ], {k, 0, n-4} ];
%Y A023426 Cf. A000108, A001006, A004148, A006318.
%Y A023426 Sequence in context: A004696 A018063 A000570 this_sequence A157134 A127926 
               A078513
%Y A023426 Adjacent sequences: A023423 A023424 A023425 this_sequence A023427 A023428 
               A023429
%K A023426 nonn,easy
%O A023426 0,5
%A A023426 Olivier Gerard (olivier.gerard(AT)gmail.com)

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