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A023426 Generalized Catalan Numbers. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

K. Park, G.S. Cheon, Lattice path counting with a bounded strip restriction

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000108, A001006, A004148, A006318.

Sequence in context: A289004 A000570 A239552 * A157134 A127926 A078513

Adjacent sequences:  A023423 A023424 A023425 * A023427 A023428 A023429

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified February 23 21:23 EST 2018. Contains 299588 sequences. (Running on oeis4.)