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A023426
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Generalized Catalan Numbers.
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1
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1, 1, 1, 1, 2, 4, 7, 11, 18, 32, 59, 107, 191, 343, 627, 1159, 2146, 3972, 7373, 13757, 25781, 48437, 91165, 171945, 325096, 616066, 1169667, 2224355, 4236728, 8082374, 15441719, 29542411, 56590472, 108532322, 208387711, 400551615
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OFFSET
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0,5
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COMMENTS
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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, Dec 23 2003
Hankel transform is A132380(n+3). [From Paul Barry, May 22 2009]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
K. Park, G.S. Cheon, Lattice path counting with a bounded strip restriction
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FORMULA
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G.f.=[1-z-sqrt((1-z)^2-4z^4)]/[2z^4]. - Emeric Deutsch, Dec 23 2003
Contribution from Paul Barry, May 22 2009: (Start)
G.f.: 1/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-... (continued fraction).
G.f.: (1/(1-x))c(x^4/(1-x)^2), c(x) the g.f. of A000108.
a(n)=sum{k=0..floor(n/4), C(n-2k,2k)*A000108(k)}. (End)
Conjecture: (n+4)*a(n) +(n+4)*a(n-1) -(5*n+8)*a(n-2) +3*n*a(n-3) +4*(2-n)*a(n-4) +12*(3-n)*a(n-5)=0. - R. J. Mathar, Sep 29 2012
a(n) ~ sqrt(3) * 2^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014
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MATHEMATICA
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Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 0, n-4} ];
CoefficientList[Series[(1-x-Sqrt[(1-x)^2-4*x^4])/(2*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
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CROSSREFS
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Cf. A000108, A001006, A004148, A006318.
Sequence in context: A289004 A000570 A239552 * A157134 A127926 A078513
Adjacent sequences: A023423 A023424 A023425 * A023427 A023428 A023429
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KEYWORD
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nonn,easy
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AUTHOR
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Olivier Gérard
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STATUS
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approved
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